Lecture 1: More Linear Algebra

A

b a

Length b minor axis radius 1 Length a circle major axis

- p. 1/38 Overview

● Overview The topics will be:

More Linear Algebra

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1)

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1) ■ Analyzing Linear ODEs (Day 1 & 2)

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1) ■ Analyzing Linear ODEs (Day 1 & 2) ■ Light Intro to Non-linear Systems (Day 2)

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1) ■ Analyzing Linear ODEs (Day 1 & 2) ■ Light Intro to Non-linear Systems (Day 2)

The philosophy:

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1) ■ Analyzing Linear ODEs (Day 1 & 2) ■ Light Intro to Non-linear Systems (Day 2)

The philosophy: get as comfortable as possible with qualitative behavior of linear systems(topic 2, requiring topic 1);

- p. 2/38 Overview

● Overview The topics will be: More Linear Algebra ■ More Linear Algebra (Day 1) ■ Analyzing Linear ODEs (Day 1 & 2) ■ Light Intro to Non-linear Systems (Day 2)

The philosophy: get as comfortable as possible with qualitative behavior of linear systems(topic 2, requiring topic 1); then understand how non-linear systems can quickly differ (topic 3).

- p. 2/38 ● Overview

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity More Linear Algebra ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 3/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity represent a lot of things. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity represent a lot of things. ● Commutativity Simultaneous linear algebra. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity represent a lot of things. ● Commutativity Simultaneous linear algebra. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Operators that map vectors to vectors: ● Eigenbases ● Behavioral Eigen-Analysis n n ● Behavioral Eigen-Analysis L : R R ; given by x Ax ● Behavioral Eigen-Analysis −→ 7→ ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity represent a lot of things. ● Commutativity Simultaneous linear algebra. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Operators that map vectors to vectors: ● Eigenbases ● Behavioral Eigen-Analysis n n ● Behavioral Eigen-Analysis L : R R ; given by x Ax ● Behavioral Eigen-Analysis −→ 7→ ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Linear ODEs: ● Behavioral Eigen-Analysis ● Nilpotency dx ● Nilpotency = Ax; x(0) = x0 ● Nilpotency dt ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Matrices Represent ALOT

● Overview Matrices: More Linear Algebra 11 12 1 ● Matrices Represent ALOT a a ... a n ● The Main Point ● a21 ... a2 A Practical Problem  n  ● Commutativity A = . . ● Commutativity . . ● Commutativity  . .  ● Rotation Matrices   ● Rotation Matrices an1 ... ann ● Commutativity   ● Commutativity   ● Commutativity represent a lot of things. ● Commutativity Simultaneous linear algebra. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Operators that map vectors to vectors: ● Eigenbases ● Behavioral Eigen-Analysis n n ● Behavioral Eigen-Analysis L : R R ; given by x Ax ● Behavioral Eigen-Analysis −→ 7→ ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Linear ODEs: ● Behavioral Eigen-Analysis ● Nilpotency dx ● Nilpotency = Ax; x(0) = x0 ● Nilpotency dt ● Nilpotency ● A Simple Form And, as you’ll see in SB200, probabilistic processes. ● A Simple Form ● A Simple Form ● A Simple Form - p. 4/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ■ ● Commutativity The range and behavior of the linear operator L ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ■ ● Commutativity The range and behavior of the linear operator L ● Commutativity ■ ● Commutativity the dynamic and steady-state behavior of x˙ = Ax ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ■ ● Commutativity The range and behavior of the linear operator L ● Commutativity ■ ● Commutativity the dynamic and steady-state behavior of x˙ = Ax ● Eigenbases ● Eigenbases ■ and a great many other things, ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ■ ● Commutativity The range and behavior of the linear operator L ● Commutativity ■ ● Commutativity the dynamic and steady-state behavior of x˙ = Ax ● Eigenbases ● Eigenbases ■ and a great many other things, ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis COMPLETELY obvious. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Main Point

● Overview But the Main Point is that:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Matrices are COMPLETELY classifiable. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity Meaning, there is a standard “view" that every matrix can be ● Rotation Matrices ● Rotation Matrices put into that renders all of its properties, like: ● Commutativity ■ ● Commutativity The existence and uniqueness of solutions to Ax = b ● Commutativity ■ ● Commutativity The range and behavior of the linear operator L ● Commutativity ■ ● Commutativity the dynamic and steady-state behavior of x˙ = Ax ● Eigenbases ● Eigenbases ■ and a great many other things, ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis COMPLETELY obvious. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency Goal of this lecture: give you intuition for how this works. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 5/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Figuring out how to calculate ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Figuring out how to calculate ● A Practical Problem ● Commutativity A ● Commutativity e ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Figuring out how to calculate ● A Practical Problem ● Commutativity A ● Commutativity e ● Commutativity ● Rotation Matrices ● Rotation Matrices efficiently. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Figuring out how to calculate ● A Practical Problem ● Commutativity A ● Commutativity e ● Commutativity ● Rotation Matrices ● Rotation Matrices efficiently. ● Commutativity ● Commutativity ● Commutativity ● Commutativity This problem is inspired by ODEs. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Practical Problem

● Overview The most important practical problem in basic linear algebra is:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Figuring out how to calculate ● A Practical Problem ● Commutativity A ● Commutativity e ● Commutativity ● Rotation Matrices ● Rotation Matrices efficiently. ● Commutativity ● Commutativity ● Commutativity ● Commutativity This problem is inspired by ODEs. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases It drives all (or really, most) of the theory. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 6/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point a b = b a. ● A Practical Problem · · ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point a b = b a. ● A Practical Problem · · ● Commutativity ● Commutativity This is the commutativity property. ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point a b = b a. ● A Practical Problem · · ● Commutativity ● Commutativity This is the commutativity property. But matices are not always ● Commutativity commutative. ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point a b = b a. ● A Practical Problem · · ● Commutativity ● Commutativity This is the commutativity property. But matices are not always ● Commutativity commutative. ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity Problem 1 Find two 2x2 matrices A and B such that ● Commutativity ● Commutativity ● Commutativity AB = BA; that is, [A, B]= AB BA = 0. ● Commutativity 6 − 6 ● Eigenbases ● Eigenbases [A, B] is called the “commutator" of A and B. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview For any two real (or complex) numbers a and b, More Linear Algebra ● Matrices Represent ALOT ● The Main Point a b = b a. ● A Practical Problem · · ● Commutativity ● Commutativity This is the commutativity property. But matices are not always ● Commutativity commutative. ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity Problem 1 Find two 2x2 matrices A and B such that ● Commutativity ● Commutativity ● Commutativity AB = BA; that is, [A, B]= AB BA = 0. ● Commutativity 6 − 6 ● Eigenbases ● Eigenbases [A, B] is called the “commutator" of A and B. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis A ● Behavioral Eigen-Analysis Notice that A always commutes with e , because ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ∞ ∞ ∞ ● Behavioral Eigen-Analysis n n+1 n A A A A ● Nilpotency Ae = A = = A. ● Nilpotency n! n! n! ● Nilpotency n=0 ! n=0 i=0 ! ● Nilpotency X X X ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 7/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity If [A, B]= AB BA = 0, then ● Rotation Matrices − ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity If [A, B]= AB BA = 0, then ● Rotation Matrices − ● Rotation Matrices A+B A B B A ● Commutativity e = e e = e e ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity If [A, B]= AB BA = 0, then ● Rotation Matrices − ● Rotation Matrices A+B A B B A ● Commutativity e = e e = e e ● Commutativity ● Commutativity ● Commutativity which might make the computation easier. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity If [A, B]= AB BA = 0, then ● Rotation Matrices − ● Rotation Matrices A+B A B B A ● Commutativity e = e e = e e ● Commutativity ● Commutativity ● Commutativity which might make the computation easier. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Problem 2 Compute ● Eigenbases ● Behavioral Eigen-Analysis 1 0 ● Behavioral Eigen-Analysis exp t . ● Behavioral Eigen-Analysis 2 1 ● Behavioral Eigen-Analysis " # ! ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Why do we care about commutativity?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Remember the motivating problem (exponentiation). ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity If [A, B]= AB BA = 0, then ● Rotation Matrices − ● Rotation Matrices A+B A B B A ● Commutativity e = e e = e e ● Commutativity ● Commutativity ● Commutativity which might make the computation easier. ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Problem 2 Compute ● Eigenbases ● Behavioral Eigen-Analysis 1 0 ● Behavioral Eigen-Analysis exp t . ● Behavioral Eigen-Analysis 2 1 ● Behavioral Eigen-Analysis " # ! ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency Answer: ● Nilpotency t ● Nilpotency e 0 ● Nilpotency t t . ● A Simple Form 2te e ● A Simple Form " # ● A Simple Form ● A Simple Form - p. 8/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But there’s a deeper interpretation.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 9/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But there’s a deeper interpretation.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 3 Show the following Little Fact 1: if A and B are diagonal ● A Practical Problem ● Commutativity matrices, then they commute. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 9/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But there’s a deeper interpretation.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 3 Show the following Little Fact 1: if A and B are diagonal ● A Practical Problem ● Commutativity matrices, then they commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: because diagonal matrix multiplication is just likea ● Rotation Matrices ● Commutativity parallel version of regular number multiplication, separately on ● Commutativity ● Commutativity each diagonal. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 9/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But there’s a deeper interpretation.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 3 Show the following Little Fact 1: if A and B are diagonal ● A Practical Problem ● Commutativity matrices, then they commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: because diagonal matrix multiplication is just likea ● Rotation Matrices ● Commutativity parallel version of regular number multiplication, separately on ● Commutativity ● Commutativity each diagonal. ● Commutativity ● Commutativity ● Commutativity Problem 4 Show the following Little Fact 2: diagonal matrices with all the ● Eigenbases ● Eigenbases diagonal numbers being the same commute with all matrices. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 9/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But there’s a deeper interpretation.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 3 Show the following Little Fact 1: if A and B are diagonal ● A Practical Problem ● Commutativity matrices, then they commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: because diagonal matrix multiplication is just likea ● Rotation Matrices ● Commutativity parallel version of regular number multiplication, separately on ● Commutativity ● Commutativity each diagonal. ● Commutativity ● Commutativity ● Commutativity Problem 4 Show the following Little Fact 2: diagonal matrices with all the ● Eigenbases ● Eigenbases diagonal numbers being the same commute with all matrices. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Reason: ; i.e. the identity matrix ● Behavioral Eigen-Analysis A(bI)= b(AI)= bA =(bI)A ● Behavioral Eigen-Analysis (obviously) commutes with everything. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 9/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview Before we go on, a little aside.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 10/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview Before we go on, a little aside.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 5 What does the matrix ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity ● Rotation Matrices Rθ = − ● Rotation Matrices "sin(θ) cos(θ) # ● Commutativity ● Commutativity ● Commutativity do? ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 10/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview Before we go on, a little aside.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 5 What does the matrix ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity ● Rotation Matrices Rθ = − ● Rotation Matrices "sin(θ) cos(θ) # ● Commutativity ● Commutativity ● Commutativity do? ● Commutativity ● Commutativity ● Commutativity Answer: it rotates the plane through angle θ. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 10/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview Before we go on, a little aside.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 5 What does the matrix ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity ● Rotation Matrices Rθ = − ● Rotation Matrices "sin(θ) cos(θ) # ● Commutativity ● Commutativity ● Commutativity do? ● Commutativity ● Commutativity ● Commutativity Answer: it rotates the plane through angle θ. ● Eigenbases ● Eigenbases What is its inverse? Well, ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 10/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview Before we go on, a little aside.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 5 What does the matrix ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity ● Rotation Matrices Rθ = − ● Rotation Matrices "sin(θ) cos(θ) # ● Commutativity ● Commutativity ● Commutativity do? ● Commutativity ● Commutativity ● Commutativity Answer: it rotates the plane through angle θ. ● Eigenbases ● Eigenbases What is its inverse? Well, ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis −1 cos( θ) sin( θ) ● Behavioral Eigen-Analysis (Rθ) = R−θ = − − − . ● Behavioral Eigen-Analysis sin( θ) cos( θ) ● " # Behavioral Eigen-Analysis − − ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency But remember cos( θ)= cos(θ) and sin( θ)= sin(θ), so ● Nilpotency − − − ● Nilpotency ● A Simple Form −1 cos(θ) sin(θ) ● A Simple Form R = . ● A Simple Form θ ● sin(θ) cos(θ) A Simple Form " # - p. 10/38 ● A Simple Form − ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview To rotate in three dimensions, we need three different More Linear Algebra rotations: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 11/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview To rotate in three dimensions, we need three different More Linear Algebra rotations: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem cos(θ) sin(θ) 0 1 0 0 ● Commutativity x,y − y,z ● Commutativity R = sin(θ) cos(θ) 0 ; R = 0 cos(θ) sin(θ) ● Commutativity θ   θ   ● Rotation Matrices − ● Rotation Matrices 0 0 1 0 sin(θ) cos(θ) ● Commutativity     ● Commutativity     ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 11/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview To rotate in three dimensions, we need three different More Linear Algebra rotations: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem cos(θ) sin(θ) 0 1 0 0 ● Commutativity x,y − y,z ● Commutativity R = sin(θ) cos(θ) 0 ; R = 0 cos(θ) sin(θ) ● Commutativity θ   θ   ● Rotation Matrices − ● Rotation Matrices 0 0 1 0 sin(θ) cos(θ) ● Commutativity     ● Commutativity     ● Commutativity and ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 11/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview To rotate in three dimensions, we need three different More Linear Algebra rotations: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem cos(θ) sin(θ) 0 1 0 0 ● Commutativity x,y − y,z ● Commutativity R = sin(θ) cos(θ) 0 ; R = 0 cos(θ) sin(θ) ● Commutativity θ   θ   ● Rotation Matrices − ● Rotation Matrices 0 0 1 0 sin(θ) cos(θ) ● Commutativity     ● Commutativity     ● Commutativity and ● Commutativity ● Commutativity cos(θ) 0 sin(θ) ● Commutativity x,z − ● Eigenbases Rθ = 0 1 0 . ● Eigenbases   ● Eigenbases sin(θ) 0 cos(θ) ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 11/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Rotation Matrices

● Overview To rotate in three dimensions, we need three different More Linear Algebra rotations: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem cos(θ) sin(θ) 0 1 0 0 ● Commutativity x,y − y,z ● Commutativity R = sin(θ) cos(θ) 0 ; R = 0 cos(θ) sin(θ) ● Commutativity θ   θ   ● Rotation Matrices − ● Rotation Matrices 0 0 1 0 sin(θ) cos(θ) ● Commutativity     ● Commutativity     ● Commutativity and ● Commutativity ● Commutativity cos(θ) 0 sin(θ) ● Commutativity x,z − ● Eigenbases Rθ = 0 1 0 . ● Eigenbases   ● Eigenbases sin(θ) 0 cos(θ) ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis There are higher-dimensional versions for each n. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 11/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity A stretches the x axis by factor a and the y axis by factor b, ● Commutativity making a circle into an ellipse. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity For example, if a< 1 and b> 1, then the picture is ● Commutativity ● Commutativity ● Commutativity ● Commutativity A ● Eigenbases ● Eigenbases b ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis a ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency radius 1 circle ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form Figure 1: Stretching action of a 2x2 diagonal matrix ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity Conversely, if a> 1 and b< 1, then the picture is ● Commutativity ● Commutativity ● Commutativity ● Commutativity A ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis b a ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Length b ● Behavioral Eigen-Analysis minor axis ● Behavioral Eigen-Analysis Length a ● Nilpotency radius 1 circle major axis ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form Figure 1: Stretching action of a 2x2 diagonal matrix ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity ● Commutativity Problem 6 What are the eigenvalues and eigenvectors of A? ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Back to commutativity. Let’s consider two 2x2 matrices. More Linear Algebra First, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem a 0 ● Commutativity A = ● Commutativity "0 b# ● Commutativity ● Rotation Matrices ● Rotation Matrices where a and b are real numbers. ● Commutativity ● Commutativity ● Commutativity Problem 6 What are the eigenvalues and eigenvectors of A? ● Commutativity ● Commutativity ● Commutativity Answer: and . ● Eigenbases ([1 0], a) ([0 1],b) ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 12/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Ok, so we have one matrix, A. Now for the second matrix, B. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 13/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Ok, so we have one matrix, A. Now for the second matrix, B. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 7 Find a 2x2 matrix B with eigenvalues α and β, and whose ● A Practical Problem ● Commutativity eigenvectors are rotated from x and y axis basis vectors by angle θ. Hint: ● Commutativity ● Commutativity use Rθ as a change-of-basis. ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 13/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Ok, so we have one matrix, A. Now for the second matrix, B. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 7 Find a 2x2 matrix B with eigenvalues α and β, and whose ● A Practical Problem ● Commutativity eigenvectors are rotated from x and y axis basis vectors by angle θ. Hint: ● Commutativity ● Commutativity use Rθ as a change-of-basis. ● Rotation Matrices ● Rotation Matrices ● Commutativity Answer: ● Commutativity ● Commutativity 2 2 ● Commutativity α 0 −1 αcos (θ)+ βsin (θ) (α β)sin(θ)cos(θ) ● Commutativity B = Rθ (Rθ) = −2 2 ● Commutativity 0 β (α β)sin(θ)cos(θ) αsin (θ)+ βcos (θ) ● Eigenbases " # " # ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 13/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Ok, so we have one matrix, A. Now for the second matrix, B. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 7 Find a 2x2 matrix B with eigenvalues α and β, and whose ● A Practical Problem ● Commutativity eigenvectors are rotated from x and y axis basis vectors by angle θ. Hint: ● Commutativity ● Commutativity use Rθ as a change-of-basis. ● Rotation Matrices ● Rotation Matrices ● Commutativity Answer: ● Commutativity ● Commutativity 2 2 ● Commutativity α 0 −1 αcos (θ)+ βsin (θ) (α β)sin(θ)cos(θ) ● Commutativity B = Rθ (Rθ) = −2 2 ● Commutativity 0 β (α β)sin(θ)cos(θ) αsin (θ)+ βcos (θ) ● Eigenbases " # " # ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis By construction, the eigenvectors and eigenvalues of B are ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ([cos(θ), sin(θ)] with value α) and [cos(θ + π/2), sin(θ + π/2)] ● Behavioral Eigen-Analysis with value β). ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 13/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Ok, so we have one matrix, A. Now for the second matrix, B. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 7 Find a 2x2 matrix B with eigenvalues α and β, and whose ● A Practical Problem ● Commutativity eigenvectors are rotated from x and y axis basis vectors by angle θ. Hint: ● Commutativity ● Commutativity use Rθ as a change-of-basis. ● Rotation Matrices ● Rotation Matrices ● Commutativity Answer: ● Commutativity ● Commutativity 2 2 ● Commutativity α 0 −1 αcos (θ)+ βsin (θ) (α β)sin(θ)cos(θ) ● Commutativity B = Rθ (Rθ) = −2 2 ● Commutativity 0 β (α β)sin(θ)cos(θ) αsin (θ)+ βcos (θ) ● Eigenbases " # " # ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis By construction, the eigenvectors and eigenvalues of B are ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ([cos(θ), sin(θ)] with value α) and [cos(θ + π/2), sin(θ + π/2)] ● Behavioral Eigen-Analysis with value β). ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency Question: why is the π/2 there in the second eigenvector? ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 13/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now let’s figure out when A and B commute. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 14/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now let’s figure out when A and B commute. On the one More Linear Algebra hand, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) a(α β)sin(θ)cos(θ) ● Commutativity AB = −2 2 . ● Commutativity " b(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Rotation Matrices − ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 14/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now let’s figure out when A and B commute. On the one More Linear Algebra hand, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) a(α β)sin(θ)cos(θ) ● Commutativity AB = −2 2 . ● Commutativity " b(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Rotation Matrices − ● Rotation Matrices ● Commutativity On the other hand, ● Commutativity ● Commutativity ● Commutativity 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) b(α β)sin(θ)cos(θ) ● Commutativity BA = −2 2 . ● Eigenbases " a(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 14/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now let’s figure out when A and B commute. On the one More Linear Algebra hand, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) a(α β)sin(θ)cos(θ) ● Commutativity AB = −2 2 . ● Commutativity " b(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Rotation Matrices − ● Rotation Matrices ● Commutativity On the other hand, ● Commutativity ● Commutativity ● Commutativity 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) b(α β)sin(θ)cos(θ) ● Commutativity BA = −2 2 . ● Eigenbases " a(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 8 When are AB and BA equal? ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 14/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now let’s figure out when A and B commute. On the one More Linear Algebra hand, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) a(α β)sin(θ)cos(θ) ● Commutativity AB = −2 2 . ● Commutativity " b(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Rotation Matrices − ● Rotation Matrices ● Commutativity On the other hand, ● Commutativity ● Commutativity ● Commutativity 2 2 ● Commutativity aαcos (θ)+ aβsin (θ) b(α β)sin(θ)cos(θ) ● Commutativity BA = −2 2 . ● Eigenbases " a(α β)sin(θ)cos(θ) bαsin (th)+ bβcos (θ)# ● Eigenbases − ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 8 When are AB and BA equal? ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: When 1) a = b, or 2) α = β or 3) θ = 0, π/2,π, 3π/2. ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 14/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview But now, let’s look at the cases one by one.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, A = aI2, where I2 is the 2x2 More Linear Algebra identity matrix. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, A = aI2, where I2 is the 2x2 More Linear Algebra identity matrix. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Claim: ● Commutativity ● Commutativity ● Rotation Matrices [cos(θ), sin(θ)] and [cos(θ + π/2), sin(θ + π/2)] ● Rotation Matrices ● Commutativity ● Commutativity are just as good eigenvectors for A as the original ones. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, A = aI2, where I2 is the 2x2 More Linear Algebra identity matrix. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Claim: ● Commutativity ● Commutativity ● Rotation Matrices [cos(θ), sin(θ)] and [cos(θ + π/2), sin(θ + π/2)] ● Rotation Matrices ● Commutativity ● Commutativity are just as good eigenvectors for A as the original ones. ● Commutativity ● Commutativity ● Commutativity ● Commutativity Problem 9 Why? (Don’t give a computational proof.) ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, A = aI2, where I2 is the 2x2 More Linear Algebra identity matrix. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Claim: ● Commutativity ● Commutativity ● Rotation Matrices [cos(θ), sin(θ)] and [cos(θ + π/2), sin(θ + π/2)] ● Rotation Matrices ● Commutativity ● Commutativity are just as good eigenvectors for A as the original ones. ● Commutativity ● Commutativity ● Commutativity ● Commutativity Problem 9 Why? (Don’t give a computational proof.) ● Eigenbases ● Eigenbases ● Eigenbases Answer: because the two original eigenvectors [0, 1] and [1, 0] ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis have the same eigenvalue, so linear combinations are also ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis eigenvectors. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Case 1: a = b. In that case, A = aI2, where I2 is the 2x2 More Linear Algebra identity matrix. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Claim: ● Commutativity ● Commutativity ● Rotation Matrices [cos(θ), sin(θ)] and [cos(θ + π/2), sin(θ + π/2)] ● Rotation Matrices ● Commutativity ● Commutativity are just as good eigenvectors for A as the original ones. ● Commutativity ● Commutativity ● Commutativity ● Commutativity Problem 9 Why? (Don’t give a computational proof.) ● Eigenbases ● Eigenbases ● Eigenbases Answer: because the two original eigenvectors [0, 1] and [1, 0] ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis have the same eigenvalue, so linear combinations are also ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis eigenvectors. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency Hence: Case 1) A and B have a common set of ● Nilpotency ⇒ ● Nilpotency eigenvectors. ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 15/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity But that means we’re in the same situation as Case 1. Again, ● Commutativity ● Commutativity common eigenvectors. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity But that means we’re in the same situation as Case 1. Again, ● Commutativity ● Commutativity common eigenvectors. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases Finally, Case 3: θ = 0, π/2,π, or 3π/2. ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity But that means we’re in the same situation as Case 1. Again, ● Commutativity ● Commutativity common eigenvectors. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases Finally, Case 3: θ = 0, π/2,π, or 3π/2. ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 10 What are the eigenvectors of in this case? ● Behavioral Eigen-Analysis B ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity But that means we’re in the same situation as Case 1. Again, ● Commutativity ● Commutativity common eigenvectors. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases Finally, Case 3: θ = 0, π/2,π, or 3π/2. ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 10 What are the eigenvectors of in this case? ● Behavioral Eigen-Analysis B ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: [0 1] and [1 0]. ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Now, Case 2: α = β. More Linear Algebra In this case, ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity 1 0 −1 −1 ● Commutativity B = Rθα (Rθ) = αRθ(Rθ) = αI2. ● Commutativity "0 1# ● Rotation Matrices ● Rotation Matrices ● Commutativity But that means we’re in the same situation as Case 1. Again, ● Commutativity ● Commutativity common eigenvectors. ● Commutativity ● Commutativity ● Commutativity ● Eigenbases Finally, Case 3: θ = 0, π/2,π, or 3π/2. ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 10 What are the eigenvectors of in this case? ● Behavioral Eigen-Analysis B ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: [0 1] and [1 0]. ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency Again! A and B have a common set of eigenvectors. ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 16/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ■ −1 −1 ● Commutativity = XDADBX since X X = I ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ■ −1 −1 ● Commutativity = XDADBX since X X = I ● Eigenbases ■ −1 ● Eigenbases = XDBDAX by Litte Fact 1 ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ■ −1 −1 ● Commutativity = XDADBX since X X = I ● Eigenbases ■ −1 ● Eigenbases = XDBDAX by Litte Fact 1 ● Eigenbases ■ −1 −1 −1 ● Behavioral Eigen-Analysis = XDB(X X)DAX – we’ve inserted X X = I ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ■ −1 −1 ● Commutativity = XDADBX since X X = I ● Eigenbases ■ −1 ● Eigenbases = XDBDAX by Litte Fact 1 ● Eigenbases ■ −1 −1 −1 ● Behavioral Eigen-Analysis = XDB(X X)DAX – we’ve inserted X X = I ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ −1 −1 ● Behavioral Eigen-Analysis =(XDBX (XDAX )= BA. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Commutativity

● Overview Little Facts 1 and 2 above also say the same thing.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 11 Show: if two matrices A and B can be diagonalized by the ● A Practical Problem ● Commutativity same matrix S, then A and B commute. ● Commutativity ● Commutativity ● Rotation Matrices Reason: ● Rotation Matrices ■ −1 −1 ● Commutativity AB =(XDAX )(XDBX ), ● Commutativity − − ● Commutativity ■ 1 1 = XDA(X X)DBX by associativity ● Commutativity ● Commutativity ■ −1 −1 ● Commutativity = XDADBX since X X = I ● Eigenbases ■ −1 ● Eigenbases = XDBDAX by Litte Fact 1 ● Eigenbases ■ −1 −1 −1 ● Behavioral Eigen-Analysis = XDB(X X)DAX – we’ve inserted X X = I ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ −1 −1 ● Behavioral Eigen-Analysis =(XDBX (XDAX )= BA. ● Behavioral Eigen-Analysis A key fact is that the converse is true. ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency Theorem 1 If A and B are both diagonalizable, then they are commutative ● Nilpotency ● A Simple Form if and only if they have a common eigenbasis. ● A Simple Form ● A Simple Form ● A Simple Form - p. 17/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point a set v1,...,vn of distinct eigenvectors. ● A Practical Problem { } ● Commutativity An eigenbasis exists IFF a matrix is diagonalizable. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point a set v1,...,vn of distinct eigenvectors. ● A Practical Problem { } ● Commutativity An eigenbasis exists IFF a matrix is diagonalizable. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity λ1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ2 0 ... 0 ● Commutativity   −1 ● Commutativity A =[v1 v2 ... vn] . . [v1 v2 ... vn] ● | | . . | | Commutativity  . .  ● Eigenbases   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point a set v1,...,vn of distinct eigenvectors. ● A Practical Problem { } ● Commutativity An eigenbasis exists IFF a matrix is diagonalizable. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity λ1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ2 0 ... 0 ● Commutativity   −1 ● Commutativity A =[v1 v2 ... vn] . . [v1 v2 ... vn] ● | | . . | | Commutativity  . .  ● Eigenbases   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis where Avi = λivi. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point a set v1,...,vn of distinct eigenvectors. ● A Practical Problem { } ● Commutativity An eigenbasis exists IFF a matrix is diagonalizable. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity λ1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ2 0 ... 0 ● Commutativity   −1 ● Commutativity A =[v1 v2 ... vn] . . [v1 v2 ... vn] ● | | . . | | Commutativity  . .  ● Eigenbases   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis where Avi = λivi. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Of course, the vi and λi might (have to) be complex, even if A ● Nilpotency ● Nilpotency is real. ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Let’s go back to the notion of an eigenbasis, that is,

More Linear Algebra ● Matrices Represent ALOT ● The Main Point a set v1,...,vn of distinct eigenvectors. ● A Practical Problem { } ● Commutativity An eigenbasis exists IFF a matrix is diagonalizable. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity λ1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ2 0 ... 0 ● Commutativity   −1 ● Commutativity A =[v1 v2 ... vn] . . [v1 v2 ... vn] ● | | . . | | Commutativity  . .  ● Eigenbases   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis where Avi = λivi. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Of course, the vi and λi might (have to) be complex, even if A ● Nilpotency ● Nilpotency is real. We’ll come back to this. ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 18/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity 1 2 n1 ● Commutativity vλ1 , vλ1 ,...,vλ1 have eigenvalue λ1 = λmax ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity 1 2 n1 ● Commutativity vλ1 , vλ1 ,...,vλ1 have eigenvalue λ1 = λmax ● Commutativity ● Commutativity 1 2 n2 ● Commutativity v , v ,...,v have eigenvalue λ2 < λ1 ● Commutativity λ2 λ2 λ2 ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity 1 2 n1 ● Commutativity vλ1 , vλ1 ,...,vλ1 have eigenvalue λ1 = λmax ● Commutativity ● Commutativity 1 2 n2 ● Commutativity v , v ,...,v have eigenvalue λ2 < λ1 ● Commutativity λ2 λ2 λ2 ● Eigenbases ● Eigenbases ● Eigenbases .... ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity 1 2 n1 ● Commutativity vλ1 , vλ1 ,...,vλ1 have eigenvalue λ1 = λmax ● Commutativity ● Commutativity 1 2 n2 ● Commutativity v , v ,...,v have eigenvalue λ2 < λ1 ● Commutativity λ2 λ2 λ2 ● Eigenbases ● Eigenbases ● Eigenbases .... ● Behavioral Eigen-Analysis m 2 nm ● Behavioral Eigen-Analysis vλm , v λm,...,vλm have eigenvalue λm = λmin. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview Now, suppose that A is diagonalizable, and has v1,...,vn { } More Linear Algebra as n = dim(A) independent eigenvectors. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Collect them in groups of equal eigenvalues, say, in decreasing ● Commutativity ● Commutativity order: ● Rotation Matrices ● Rotation Matrices ● Commutativity 1 2 n1 ● Commutativity vλ1 , vλ1 ,...,vλ1 have eigenvalue λ1 = λmax ● Commutativity ● Commutativity 1 2 n2 ● Commutativity v , v ,...,v have eigenvalue λ2 < λ1 ● Commutativity λ2 λ2 λ2 ● Eigenbases ● Eigenbases ● Eigenbases .... ● Behavioral Eigen-Analysis m 2 nm ● Behavioral Eigen-Analysis vλm , v λm,...,vλm have eigenvalue λm = λmin. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis m = number of distinct eigenvalues, and ● Nilpotency n1 + n2 + ... + nm = ni = dim(A)= n. ● Nilpotency i ● Nilpotency ● Nilpotency P ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 19/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point λ1In1 0 0 ... 0 ● A Practical Problem ● Commutativity 0 λ2In2 0 ... 0 ● Commutativity   ● Commutativity A = . . . ● Rotation Matrices . . ● Rotation Matrices   ● Commutativity   ● Commutativity  0 0 ... 0 λmInm  ● Commutativity   ● Commutativity   ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point λ1In1 0 0 ... 0 ● A Practical Problem ● Commutativity 0 λ2In2 0 ... 0 ● Commutativity   ● Commutativity A = . . . ● Rotation Matrices . . ● Rotation Matrices   ● Commutativity   ● Commutativity  0 0 ... 0 λmInm  ● Commutativity   ● Commutativity Each block is the eigenspace associated with eigenvalue ● Commutativity λiIni ● Commutativity λi. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point λ1In1 0 0 ... 0 ● A Practical Problem ● Commutativity 0 λ2In2 0 ... 0 ● Commutativity   ● Commutativity A = . . . ● Rotation Matrices . . ● Rotation Matrices   ● Commutativity   ● Commutativity  0 0 ... 0 λmInm  ● Commutativity   ● Commutativity Each block is the eigenspace associated with eigenvalue ● Commutativity λiIni ● Commutativity λi. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 12 Show: for each i, Aλi is a vector space with dimension ni. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point λ1In1 0 0 ... 0 ● A Practical Problem ● Commutativity 0 λ2In2 0 ... 0 ● Commutativity   ● Commutativity A = . . . ● Rotation Matrices . . ● Rotation Matrices   ● Commutativity   ● Commutativity  0 0 ... 0 λmInm  ● Commutativity   ● Commutativity Each block is the eigenspace associated with eigenvalue ● Commutativity λiIni ● Commutativity λi. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 12 Show: for each i, Aλi is a vector space with dimension ni. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Reason: linear combinations of eigenvectors with the same ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis eigenvalue are also eigenvectors with that eigenvalue. ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Eigenbases

● Overview We can think of A (after some change of basis) as More Linear Algebra ● Matrices Represent ALOT ● The Main Point λ1In1 0 0 ... 0 ● A Practical Problem ● Commutativity 0 λ2In2 0 ... 0 ● Commutativity   ● Commutativity A = . . . ● Rotation Matrices . . ● Rotation Matrices   ● Commutativity   ● Commutativity  0 0 ... 0 λmInm  ● Commutativity   ● Commutativity Each block is the eigenspace associated with eigenvalue ● Commutativity λiIni ● Commutativity λi. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis Problem 12 Show: for each i, Aλi is a vector space with dimension ni. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Reason: linear combinations of eigenvectors with the same ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis eigenvalue are also eigenvectors with that eigenvalue. ● Nilpotency ● Nilpotency For the same reason, any basis of Aλi is equivalent to any ● Nilpotency ● Nilpotency other, for the purposes of diagonalization. ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 20/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s get back to the issue of complex and real eigenvalues.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 21/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s get back to the issue of complex and real eigenvalues. More Linear Algebra Suppose all the eigenvalues of A are real. Then ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 21/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s get back to the issue of complex and real eigenvalues. More Linear Algebra Suppose all the eigenvalues of A are real. Then ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity A = SDS where ● Commutativity ● Commutativity ● Rotation Matrices λ1 0 0 ... 0 ● Rotation Matrices ● Commutativity 0 λ2 0 ... 0 ● Commutativity   ● D = Commutativity . . ● Commutativity . . ● Commutativity   ● Commutativity   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis is a diagonal matrix with all real entries along the diagonal. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 21/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s get back to the issue of complex and real eigenvalues. More Linear Algebra Suppose all the eigenvalues of A are real. Then ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity A = SDS where ● Commutativity ● Commutativity ● Rotation Matrices λ1 0 0 ... 0 ● Rotation Matrices ● Commutativity 0 λ2 0 ... 0 ● Commutativity   ● D = Commutativity . . ● Commutativity . . ● Commutativity   ● Commutativity   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis is a diagonal matrix with all real entries along the diagonal. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Question: what does a diagonal matrix with real entries ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis correspond to? (don’t forget some could be negative) ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 21/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s get back to the issue of complex and real eigenvalues. More Linear Algebra Suppose all the eigenvalues of A are real. Then ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity A = SDS where ● Commutativity ● Commutativity ● Rotation Matrices λ1 0 0 ... 0 ● Rotation Matrices ● Commutativity 0 λ2 0 ... 0 ● Commutativity   ● D = Commutativity . . ● Commutativity . . ● Commutativity   ● Commutativity   ● Eigenbases  0 0 ... 0 λn ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis is a diagonal matrix with all real entries along the diagonal. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Question: what does a diagonal matrix with real entries ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis correspond to? (don’t forget some could be negative) ● Nilpotency ● Nilpotency Answer: stretching along various directions, with a flip as well if ● Nilpotency ● Nilpotency negative. ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 21/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity Answer: v1 = [1 i], with eigenvalue cos(θ) isin(θ) and ● Commutativity − ● Commutativity v2 =[i 1] with eigenvalue cos(θ)+ isin(θ). ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity Answer: v1 = [1 i], with eigenvalue cos(θ) isin(θ) and ● Commutativity − ● Commutativity v2 =[i 1] with eigenvalue cos(θ)+ isin(θ). ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Now, notice three facts: ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity Answer: v1 = [1 i], with eigenvalue cos(θ) isin(θ) and ● Commutativity − ● Commutativity v2 =[i 1] with eigenvalue cos(θ)+ isin(θ). ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Now, notice three facts: ● Eigenbases ■ ● Behavioral Eigen-Analysis Rotation here corresponds to complex ● Behavioral Eigen-Analysis eigenvalues/eigenvectors. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity Answer: v1 = [1 i], with eigenvalue cos(θ) isin(θ) and ● Commutativity − ● Commutativity v2 =[i 1] with eigenvalue cos(θ)+ isin(θ). ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Now, notice three facts: ● Eigenbases ■ ● Behavioral Eigen-Analysis Rotation here corresponds to complex ● Behavioral Eigen-Analysis eigenvalues/eigenvectors. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Behavioral Eigen-Analysis The eigenvalues are complex conjugates of each other. ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hm, what does a complex eigenvalue correspond to?

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 13 Compute the eigenvalues and eigenvectors of ● A Practical Problem ● Commutativity ● Commutativity cos(θ) sin(θ) ● Commutativity Rθ = − . ● Rotation Matrices "sin(θ) cos(θ) # ● Rotation Matrices ● Commutativity ● Commutativity Answer: v1 = [1 i], with eigenvalue cos(θ) isin(θ) and ● Commutativity − ● Commutativity v2 =[i 1] with eigenvalue cos(θ)+ isin(θ). ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases Now, notice three facts: ● Eigenbases ■ ● Behavioral Eigen-Analysis Rotation here corresponds to complex ● Behavioral Eigen-Analysis eigenvalues/eigenvectors. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Behavioral Eigen-Analysis The eigenvalues are complex conjugates of each other. ● Behavioral Eigen-Analysis ■ ● Nilpotency Conjugate pair corresponds to real 2x2 with equal diagonal ● Nilpotency ● Nilpotency elements, off-diagonals. ● Nilpotency ± ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 22/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ■ If Ax = λx, then conjugating gives: ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ■ If Ax = λx, then conjugating gives: ● Rotation Matrices ● Commutativity ■ ¯ ¯ ● Commutativity Ax = λx = Ax¯ = λx¯. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ■ If Ax = λx, then conjugating gives: ● Rotation Matrices ● Commutativity ■ ¯ ¯ ● Commutativity Ax = λx = Ax¯ = λx¯. ● Commutativity ■ ¯ ● Commutativity since A is real, A = A, so ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ■ If Ax = λx, then conjugating gives: ● Rotation Matrices ● Commutativity ■ ¯ ¯ ● Commutativity Ax = λx = Ax¯ = λx¯. ● Commutativity ■ ¯ ● Commutativity since A is real, A = A, so ● Commutativity ■ ● Commutativity Ax¯ = λ¯x¯ ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Let’s see why these facts are general.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 14 Show that of λ is an eigenvalue of a real matrix, so is λ¯. ● A Practical Problem ● Commutativity ● Commutativity Reason: ● Commutativity ● Rotation Matrices ■ If Ax = λx, then conjugating gives: ● Rotation Matrices ● Commutativity ■ ¯ ¯ ● Commutativity Ax = λx = Ax¯ = λx¯. ● Commutativity ■ ¯ ● Commutativity since A is real, A = A, so ● Commutativity ■ ● Commutativity Ax¯ = λ¯x¯ ● Eigenbases ● Eigenbases ■ so λ¯ is an eigenvalue by definition. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 23/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hence, the eigenvalues of A can be listed in two groups. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 24/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hence, the eigenvalues of A can be listed in two groups. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Real values: r1,r2,...,rk and ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 24/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hence, the eigenvalues of A can be listed in two groups. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Real values: r1,r2,...,rk and non-real values: ● A Practical Problem ● Commutativity c1, c¯1,c2, c¯2,...,cl, c¯l. ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 24/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hence, the eigenvalues of A can be listed in two groups. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Real values: r1,r2,...,rk and non-real values: ● A Practical Problem ● Commutativity c1, c¯1,c2, c¯2,...,cl, c¯l. ● Commutativity ● Commutativity So, when diagonalized, ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 24/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Hence, the eigenvalues of A can be listed in two groups. More Linear Algebra ● Matrices Represent ALOT ● The Main Point Real values: r1,r2,...,rk and non-real values: ● A Practical Problem ● Commutativity c1, c¯1,c2, c¯2,...,cl, c¯l. ● Commutativity ● Commutativity So, when diagonalized, ● Rotation Matrices ● Rotation Matrices ● Commutativity r1 0 0 0 ... 0 0 0 ● Commutativity ● Commutativity 0 r2 0 0 ... 0 0 0 ● Commutativity   ● Commutativity . . ● Commutativity . . ● Eigenbases  0 0 ... 0 0  ● Eigenbases   ●  0 ... r 0 ... 0 0  Eigenbases  k  ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis A =  0 0 0 c1 0 ... 0 0 . ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis  0000c ¯1 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis  . .  ● Nilpotency  . .  ● Nilpotency   ● Nilpotency  0 cl 0 ● Nilpotency   ● A Simple Form   ● A Simple Form  0 0c ¯l ● A Simple Form   ● A Simple Form   - p. 24/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices In fact, they correspond to real matrices. Namely: ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices In fact, they correspond to real matrices. Namely: Blocki is the ● Commutativity same up to change of basis as ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices In fact, they correspond to real matrices. Namely: Blocki is the ● Commutativity same up to change of basis as ● Commutativity ● Commutativity ● Commutativity ● Commutativity ai bi ● Commutativity − ● Eigenbases "bi ai # ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis where a = Re(c ) and b = Im(c ). ● Behavioral Eigen-Analysis i i i i ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices In fact, they correspond to real matrices. Namely: Blocki is the ● Commutativity same up to change of basis as ● Commutativity ● Commutativity ● Commutativity ● Commutativity ai bi ● Commutativity − ● Eigenbases "bi ai # ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis where a = Re(c ) and b = Im(c ). ● Behavioral Eigen-Analysis i i i i ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 15 This is a constant times a rotation matrix. Which one? (Hint: ● Behavioral Eigen-Analysis ● Nilpotency use the definition of cosine.) ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview But consider those complex pair-blocks

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ci 0 ● A Practical Problem Blocki = . ● Commutativity 0c ¯i ● Commutativity " # ● Commutativity ● Rotation Matrices ● Rotation Matrices In fact, they correspond to real matrices. Namely: Blocki is the ● Commutativity same up to change of basis as ● Commutativity ● Commutativity ● Commutativity ● Commutativity ai bi ● Commutativity − ● Eigenbases "bi ai # ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis where a = Re(c ) and b = Im(c ). ● Behavioral Eigen-Analysis i i i i ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 15 This is a constant times a rotation matrix. Which one? (Hint: ● Behavioral Eigen-Analysis ● Nilpotency use the definition of cosine.) ● Nilpotency ● Nilpotency −1 2 2 ● Nilpotency Answer: rotation angle is θi = cos (ai/ a + b ) ● A Simple Form i i ● A Simple Form ● A Simple Form p ● A Simple Form - p. 25/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Thus, we have a TOTAL behavioral understanding of More Linear Algebra diagonalizable real matrices: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 26/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Thus, we have a TOTAL behavioral understanding of More Linear Algebra diagonalizable real matrices: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity Theorem 2 All such matrices can be written as SDS where D has ● Commutativity ● Commutativity diagonal elements corresponding to real-eigenvalue dilations or 2x2 blocks ● Rotation Matrices ● Rotation Matrices corresponding to complex-eigenvalue rotation-dilations. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 26/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Thus, we have a TOTAL behavioral understanding of More Linear Algebra diagonalizable real matrices: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity Theorem 2 All such matrices can be written as SDS where D has ● Commutativity ● Commutativity diagonal elements corresponding to real-eigenvalue dilations or 2x2 blocks ● Rotation Matrices ● Rotation Matrices corresponding to complex-eigenvalue rotation-dilations. ● Commutativity ● Commutativity ● Commutativity Moreover: the rotation rate and stretch multiple are controlled ● Commutativity ● Commutativity by the eigenvalues. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 26/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Thus, we have a TOTAL behavioral understanding of More Linear Algebra diagonalizable real matrices: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity Theorem 2 All such matrices can be written as SDS where D has ● Commutativity ● Commutativity diagonal elements corresponding to real-eigenvalue dilations or 2x2 blocks ● Rotation Matrices ● Rotation Matrices corresponding to complex-eigenvalue rotation-dilations. ● Commutativity ● Commutativity ● Commutativity Moreover: the rotation rate and stretch multiple are controlled ● Commutativity ● Commutativity by the eigenvalues. ● Commutativity ● Eigenbases But what if the matrix is not diagonalizable? i.e ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 26/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Behavioral Eigen-Analysis

● Overview Thus, we have a TOTAL behavioral understanding of More Linear Algebra diagonalizable real matrices: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem −1 ● Commutativity Theorem 2 All such matrices can be written as SDS where D has ● Commutativity ● Commutativity diagonal elements corresponding to real-eigenvalue dilations or 2x2 blocks ● Rotation Matrices ● Rotation Matrices corresponding to complex-eigenvalue rotation-dilations. ● Commutativity ● Commutativity ● Commutativity Moreover: the rotation rate and stretch multiple are controlled ● Commutativity ● Commutativity by the eigenvalues. ● Commutativity ● Eigenbases But what if the matrix is not diagonalizable? i.e ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis What if we can’t find n linearly independent eigenvectors? ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 26/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem k ● Commutativity For all real (or complex) numbers a, if a = 0 a = 0. ● Commutativity ⇒ ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem k ● Commutativity For all real (or complex) numbers a, if a = 0 a = 0. ● Commutativity ⇒ ● Commutativity ● Rotation Matrices 2 ● Rotation Matrices Problem 16 Find a non-zero 2x2 matrix A such that A = 0. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem k ● Commutativity For all real (or complex) numbers a, if a = 0 a = 0. ● Commutativity ⇒ ● Commutativity ● Rotation Matrices 2 ● Rotation Matrices Problem 16 Find a non-zero 2x2 matrix A such that A = 0. ● Commutativity ● Commutativity ● Commutativity Answer: the standard answer is ● Commutativity ● Commutativity ● Commutativity 0 1 ● Eigenbases A = . ● Eigenbases 0 0 ● Eigenbases " # ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem k ● Commutativity For all real (or complex) numbers a, if a = 0 a = 0. ● Commutativity ⇒ ● Commutativity ● Rotation Matrices 2 ● Rotation Matrices Problem 16 Find a non-zero 2x2 matrix A such that A = 0. ● Commutativity ● Commutativity ● Commutativity Answer: the standard answer is ● Commutativity ● Commutativity ● Commutativity 0 1 ● Eigenbases A = . ● Eigenbases 0 0 ● Eigenbases " # ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 17 Compute the eigenvalues and eigenvalues of this A. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview We started with one property of scalars that matrices didn’t More Linear Algebra share (commutativity). Here’s another: ● Matrices Represent ALOT ● The Main Point ● A Practical Problem k ● Commutativity For all real (or complex) numbers a, if a = 0 a = 0. ● Commutativity ⇒ ● Commutativity ● Rotation Matrices 2 ● Rotation Matrices Problem 16 Find a non-zero 2x2 matrix A such that A = 0. ● Commutativity ● Commutativity ● Commutativity Answer: the standard answer is ● Commutativity ● Commutativity ● Commutativity 0 1 ● Eigenbases A = . ● Eigenbases 0 0 ● Eigenbases " # ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 17 Compute the eigenvalues and eigenvalues of this A. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: Trick question. A has no non-trivial eigenvectors. ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 27/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT −1 ● The Main Point Problem 18 Find an n-by-n matrix such that An = 0 but Ai = 0 for ● A Practical Problem 6 ● Commutativity i < n 1 (A is the said to be “nilpotent of order n 1”). ● Commutativity − − ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT −1 ● The Main Point Problem 18 Find an n-by-n matrix such that An = 0 but Ai = 0 for ● A Practical Problem 6 ● Commutativity i < n 1 (A is the said to be “nilpotent of order n 1”). ● Commutativity − − ● Commutativity ● Rotation Matrices Answer: Standard answer is ● Rotation Matrices ● Commutativity ● Commutativity 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . ● Eigenbases Nn = . . . ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 1 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT −1 ● The Main Point Problem 18 Find an n-by-n matrix such that An = 0 but Ai = 0 for ● A Practical Problem 6 ● Commutativity i < n 1 (A is the said to be “nilpotent of order n 1”). ● Commutativity − − ● Commutativity ● Rotation Matrices Answer: Standard answer is ● Rotation Matrices ● Commutativity ● Commutativity 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . ● Eigenbases Nn = . . . ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 1 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis This matrix has no non-trivial eigenvectors. ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT −1 ● The Main Point Problem 18 Find an n-by-n matrix such that An = 0 but Ai = 0 for ● A Practical Problem 6 ● Commutativity i < n 1 (A is the said to be “nilpotent of order n 1”). ● Commutativity − − ● Commutativity ● Rotation Matrices Answer: Standard answer is ● Rotation Matrices ● Commutativity ● Commutativity 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . ● Eigenbases Nn = . . . ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 1 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis This matrix has no non-trivial eigenvectors. It’s behavior is like ● Nilpotency ● Nilpotency a step-by-step “collapser”: ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview A matrix is said to be nilpotent if Ak = 0 for some k. More Linear Algebra ● Matrices Represent ALOT −1 ● The Main Point Problem 18 Find an n-by-n matrix such that An = 0 but Ai = 0 for ● A Practical Problem 6 ● Commutativity i < n 1 (A is the said to be “nilpotent of order n 1”). ● Commutativity − − ● Commutativity ● Rotation Matrices Answer: Standard answer is ● Rotation Matrices ● Commutativity ● Commutativity 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . ● Eigenbases Nn = . . . ● Eigenbases   ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 1 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis This matrix has no non-trivial eigenvectors. It’s behavior is like ● Nilpotency ● Nilpotency a step-by-step “collapser”: ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form en en−1 ... e1 0. ● A Simple Form → → → → ● A Simple Form - p. 28/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ■ l l ● Commutativity Then N x = λ x for all l. ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ■ l l ● Commutativity Then N x = λ x for all l. ● Rotation Matrices ■ l ● Rotation Matrices But λ = 0 for all l, ● Commutativity 6 ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ■ l l ● Commutativity Then N x = λ x for all l. ● Rotation Matrices ■ l ● Rotation Matrices But λ = 0 for all l, ● Commutativity 6 ■ k ● Commutativity which conflicts with N = 0 for some k. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ■ l l ● Commutativity Then N x = λ x for all l. ● Rotation Matrices ■ l ● Rotation Matrices But λ = 0 for all l, ● Commutativity 6 ■ k ● Commutativity which conflicts with N = 0 for some k. ● Commutativity ● Commutativity ● Commutativity ● Commutativity So we might suspect: nilpotent matrices fill in the “hole” left by ● Eigenbases ● Eigenbases the non-diagonalizable parts of arbitrary matrix. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Problem 19 Show that any nilpotent matrix has no non-trivial eigenvectors.

More Linear Algebra ● Matrices Represent ALOT Reason: ● The Main Point ● A Practical Problem ■ Suppose Nx = λx and N is nilpotent. ● Commutativity ● Commutativity ■ l l ● Commutativity Then N x = λ x for all l. ● Rotation Matrices ■ l ● Rotation Matrices But λ = 0 for all l, ● Commutativity 6 ■ k ● Commutativity which conflicts with N = 0 for some k. ● Commutativity ● Commutativity ● Commutativity ● Commutativity So we might suspect: nilpotent matrices fill in the “hole” left by ● Eigenbases ● Eigenbases the non-diagonalizable parts of arbitrary matrix. ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis The lack of eigenvectors of nilpotent matrices could make up ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis for the missing dimensions in a non-diagonal matrix. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 29/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Theorem 3 All nilpotent matrices can be put into the standard form – that is

More Linear Algebra all zeros, except 1s on the “super-diagonal”. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 30/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Theorem 3 All nilpotent matrices can be put into the standard form – that is

More Linear Algebra all zeros, except 1s on the “super-diagonal”. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem That is, if an n-by-n matrix A is nilpotent of order n 1, then ● Commutativity − ● Commutativity there is an invertible matrix S such that ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 30/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Theorem 3 All nilpotent matrices can be put into the standard form – that is

More Linear Algebra all zeros, except 1s on the “super-diagonal”. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem That is, if an n-by-n matrix A is nilpotent of order n 1, then ● Commutativity − ● Commutativity there is an invertible matrix S such that ● Commutativity ● Rotation Matrices ● Rotation Matrices 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . −1 −1 ● . . Commutativity A = S . . S = SNnS . ● Commutativity   ● Eigenbases   ● Eigenbases 0 ... 0 1 ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 30/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Theorem 3 All nilpotent matrices can be put into the standard form – that is

More Linear Algebra all zeros, except 1s on the “super-diagonal”. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem That is, if an n-by-n matrix A is nilpotent of order n 1, then ● Commutativity − ● Commutativity there is an invertible matrix S such that ● Commutativity ● Rotation Matrices ● Rotation Matrices 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . −1 −1 ● . . Commutativity A = S . . S = SNnS . ● Commutativity   ● Eigenbases   ● Eigenbases 0 ... 0 1 ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis Lower-order nilpotency some smaller Ni blocks. ● Behavioral Eigen-Analysis ⇒ ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 30/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Nilpotency

● Overview Theorem 3 All nilpotent matrices can be put into the standard form – that is

More Linear Algebra all zeros, except 1s on the “super-diagonal”. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem That is, if an n-by-n matrix A is nilpotent of order n 1, then ● Commutativity − ● Commutativity there is an invertible matrix S such that ● Commutativity ● Rotation Matrices ● Rotation Matrices 0 1 00 ... 0 ● Commutativity ● Commutativity 0 0 10 ... 0 ● Commutativity   ● Commutativity . . −1 −1 ● . . Commutativity A = S . . S = SNnS . ● Commutativity   ● Eigenbases   ● Eigenbases 0 ... 0 1 ● Eigenbases   ● Behavioral Eigen-Analysis 0 ... 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis Lower-order nilpotency some smaller Ni blocks. ● Behavioral Eigen-Analysis ⇒ ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency To fill in the “missing eigenvector" gap, let’s add diagonal ● Nilpotency matrices to nilpontent matrices. ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 30/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Let λ be any complex number. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let λ be any complex number. Now, let Jn = λIn + Nn. More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let λ be any complex number. Now, let Jn = λIn + Nn. This More Linear Algebra ● Matrices Represent ALOT is the sum of the simplest diagonal and nilpotent matrices. ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let λ be any complex number. Now, let Jn = λIn + Nn. This More Linear Algebra ● Matrices Represent ALOT is the sum of the simplest diagonal and nilpotent matrices. ● The Main Point ● A Practical Problem ● Commutativity λ 1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ 1 0 ... 0 ● Rotation Matrices   ● Rotation Matrices λ . . ● Commutativity Jn = . . . ● Commutativity   ● Commutativity   ● Commutativity 0 ... λ 1 ● Commutativity   ● Commutativity 0 ... λ ● Eigenbases   ● Eigenbases   ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let λ be any complex number. Now, let Jn = λIn + Nn. This More Linear Algebra ● Matrices Represent ALOT is the sum of the simplest diagonal and nilpotent matrices. ● The Main Point ● A Practical Problem ● Commutativity λ 1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ 1 0 ... 0 ● Rotation Matrices   ● Rotation Matrices λ . . ● Commutativity Jn = . . . ● Commutativity   ● Commutativity   ● Commutativity 0 ... λ 1 ● Commutativity   ● Commutativity 0 ... λ ● Eigenbases   ● Eigenbases   ● Eigenbases λ ● Behavioral Eigen-Analysis Problem 20 What are the eigenvalues/vectors of Jn ? ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let λ be any complex number. Now, let Jn = λIn + Nn. This More Linear Algebra ● Matrices Represent ALOT is the sum of the simplest diagonal and nilpotent matrices. ● The Main Point ● A Practical Problem ● Commutativity λ 1 0 0 ... 0 ● Commutativity ● Commutativity 0 λ 1 0 ... 0 ● Rotation Matrices   ● Rotation Matrices λ . . ● Commutativity Jn = . . . ● Commutativity   ● Commutativity   ● Commutativity 0 ... λ 1 ● Commutativity   ● Commutativity 0 ... λ ● Eigenbases   ● Eigenbases   ● Eigenbases λ ● Behavioral Eigen-Analysis Problem 20 What are the eigenvalues/vectors of Jn ? ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: [1 0 ... 0] is the only eigenvector, with eigenvalue λ. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 31/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... but, ● The Main Point ● A Practical Problem ● Commutativity Problem 21 Compute ● Commutativity λ n−1 ● Commutativity (J λIn) . ● Rotation Matrices n ● Rotation Matrices − ● Commutativity Fast. ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... but, ● The Main Point ● A Practical Problem ● Commutativity Problem 21 Compute ● Commutativity λ n−1 ● Commutativity (J λIn) . ● Rotation Matrices n ● Rotation Matrices − ● Commutativity Fast. ● Commutativity ● Commutativity ● Commutativity Answer: 0, b/c the stuff inside the parentheses is just Nn, the ● Commutativity ● Commutativity nilpotent matrix. ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... but, ● The Main Point ● A Practical Problem ● Commutativity Problem 21 Compute ● Commutativity λ n−1 ● Commutativity (J λIn) . ● Rotation Matrices n ● Rotation Matrices − ● Commutativity Fast. ● Commutativity ● Commutativity ● Commutativity Answer: 0, b/c the stuff inside the parentheses is just Nn, the ● Commutativity ● Commutativity nilpotent matrix. ● Eigenbases ● Eigenbases ● Eigenbases Hence, all vectors are “generalized" eigenvectors; ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... but, ● The Main Point ● A Practical Problem ● Commutativity Problem 21 Compute ● Commutativity λ n−1 ● Commutativity (J λIn) . ● Rotation Matrices n ● Rotation Matrices − ● Commutativity Fast. ● Commutativity ● Commutativity ● Commutativity Answer: 0, b/c the stuff inside the parentheses is just Nn, the ● Commutativity ● Commutativity nilpotent matrix. ● Eigenbases ● Eigenbases ● Eigenbases Hence, all vectors are “generalized" eigenvectors; is a ● Behavioral Eigen-Analysis x ● Behavioral Eigen-Analysis generalized eigenvector of with generalized eigenvalue if there is a ● A λ Behavioral Eigen-Analysis k ● Behavioral Eigen-Analysis k such that (A λIn) (x) = 0. ● Behavioral Eigen-Analysis − ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Ok, [1 0 ... 0] was the only eigenvector of Jn , so it’s not More Linear Algebra ● Matrices Represent ALOT diagonalizable, etc... but, ● The Main Point ● A Practical Problem ● Commutativity Problem 21 Compute ● Commutativity λ n−1 ● Commutativity (J λIn) . ● Rotation Matrices n ● Rotation Matrices − ● Commutativity Fast. ● Commutativity ● Commutativity ● Commutativity Answer: 0, b/c the stuff inside the parentheses is just Nn, the ● Commutativity ● Commutativity nilpotent matrix. ● Eigenbases ● Eigenbases ● Eigenbases Hence, all vectors are “generalized" eigenvectors; is a ● Behavioral Eigen-Analysis x ● Behavioral Eigen-Analysis generalized eigenvector of with generalized eigenvalue if there is a ● A λ Behavioral Eigen-Analysis k ● Behavioral Eigen-Analysis k such that (A λIn) (x) = 0. ● Behavioral Eigen-Analysis − ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency (.... it a power of k where there was 1 in the original definition) ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 32/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let’s investigate the behavior of Jn . More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 33/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let’s investigate the behavior of Jn . More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 22 Compute ● A Practical Problem ● Commutativity ● Commutativity l ● Commutativity 0 λ 1 0 0 ● Rotation Matrices λ l ● Rotation Matrices (J3 ) 1 = 0 λ 1 1 . ● Commutativity       ● Commutativity ● Commutativity 0 0 0 λ 0 ● Commutativity       ● Commutativity       ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 33/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let’s investigate the behavior of Jn . More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 22 Compute ● A Practical Problem ● Commutativity ● Commutativity l ● Commutativity 0 λ 1 0 0 ● Rotation Matrices λ l ● Rotation Matrices (J3 ) 1 = 0 λ 1 1 . ● Commutativity       ● Commutativity ● Commutativity 0 0 0 λ 0 ● Commutativity       ● Commutativity       ● Commutativity Answer: ● Eigenbases l−1 ● Eigenbases lλ ● Eigenbases l ● Behavioral Eigen-Analysis λ . ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 33/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview λ Let’s investigate the behavior of Jn . More Linear Algebra ● Matrices Represent ALOT ● The Main Point Problem 22 Compute ● A Practical Problem ● Commutativity ● Commutativity l ● Commutativity 0 λ 1 0 0 ● Rotation Matrices λ l ● Rotation Matrices (J3 ) 1 = 0 λ 1 1 . ● Commutativity       ● Commutativity ● Commutativity 0 0 0 λ 0 ● Commutativity       ● Commutativity       ● Commutativity Answer: ● Eigenbases l−1 ● Eigenbases lλ ● Eigenbases l ● Behavioral Eigen-Analysis λ . ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis 0 ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis (Aside: doesn’t it remind you of derivatives?) ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 33/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Pictorially:

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity [0 1 0] ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT ● The Main Point [1 λ 0] ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT 2 ● The Main Point [2λ λ 0] ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity [n λn−1 λ n 0] ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity [n λn−1 λ n 0] ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 23 What angle does this make with the x-axis, as l ? ● Behavioral Eigen-Analysis → ∞ ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview y−axis More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity [n λn−1 λ n 0] ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity x−axis ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Problem 23 What angle does this make with the x-axis, as l ? ● Behavioral Eigen-Analysis → ∞ ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Answer: ● Behavioral Eigen-Analysis ● Nilpotency n−1 ● Nilpotency −1 nλ n −1 ● Nilpotency lim cos = = cos (1) = 0. ● − 2 2 Nilpotency l→∞ (nλn 1)2 +(λn)2 √n + λ ● A Simple Form ! ● A Simple Form ● A Simple Form p ● A Simple Form - p. 34/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Now, if you compute

More Linear Algebra ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 35/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Now, if you compute

More Linear Algebra ● Matrices Represent ALOT l ● The Main Point ● A Practical Problem 0 λ 1 0 0 ● Commutativity λ l ● Commutativity (J3 ) 0 = 0 λ 1 0 , ● Commutativity       ● Rotation Matrices 1 0 0 λ 1 ● Rotation Matrices       ● Commutativity       ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 35/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Now, if you compute

More Linear Algebra ● Matrices Represent ALOT l ● The Main Point ● A Practical Problem 0 λ 1 0 0 ● Commutativity λ l ● Commutativity (J3 ) 0 = 0 λ 1 0 , ● Commutativity       ● Rotation Matrices 1 0 0 λ 1 ● Rotation Matrices       ● Commutativity       ● Commutativity You get ● Commutativity l−2 ● Commutativity (l(l 1)/2)λ 1 ● Commutativity − l−1 λ ● Commutativity lλ O( ) . ● n Eigenbases   ∝  2  ● Eigenbases l λ 2 ● Eigenbases λ O( n ) ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 35/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Now, if you compute

More Linear Algebra ● Matrices Represent ALOT l ● The Main Point ● A Practical Problem 0 λ 1 0 0 ● Commutativity λ l ● Commutativity (J3 ) 0 = 0 λ 1 0 , ● Commutativity       ● Rotation Matrices 1 0 0 λ 1 ● Rotation Matrices       ● Commutativity       ● Commutativity You get ● Commutativity l−2 ● Commutativity (l(l 1)/2)λ 1 ● Commutativity − l−1 λ ● Commutativity lλ O( ) . ● n Eigenbases   ∝  2  ● Eigenbases l λ 2 ● Eigenbases λ O( n ) ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Just as above, as l , the angle with x-axis moves toward ● Behavioral Eigen-Analysis → ∞ ● Nilpotency zero. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 35/38 ● A Simple Form ● Putting it All Together ● Putting it All Together A Simple Form

● Overview Now, if you compute

More Linear Algebra ● Matrices Represent ALOT l ● The Main Point ● A Practical Problem 0 λ 1 0 0 ● Commutativity λ l ● Commutativity (J3 ) 0 = 0 λ 1 0 , ● Commutativity       ● Rotation Matrices 1 0 0 λ 1 ● Rotation Matrices       ● Commutativity       ● Commutativity You get ● Commutativity l−2 ● Commutativity (l(l 1)/2)λ 1 ● Commutativity − l−1 λ ● Commutativity lλ O( ) . ● n Eigenbases   ∝  2  ● Eigenbases l λ 2 ● Eigenbases λ O( n ) ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis     ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Just as above, as l , the angle with x-axis moves toward ● Behavioral Eigen-Analysis →λ ∞ ● Nilpotency zero. Conclusion: Jn “pushes" all the generalized ● Nilpotency ● Nilpotency eigenvectors down (asymptotically) to a true eigenvector. ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 35/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview The most celebrated and powerful result of linear algebra is More Linear Algebra now in reach. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity ● Commutativity ● Commutativity ● Rotation Matrices ● Rotation Matrices ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 36/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview The most celebrated and powerful result of linear algebra is More Linear Algebra now in reach. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Theorem 4 (The Jordan Normal Form) Given any matrix A, there is an ● Commutativity −1 ● Commutativity invertible matrix S such that A = SDS , where ● Rotation Matrices ● Rotation Matrices λ1 ● Commutativity Jn1 0 0 ... 0 ● Commutativity ● Commutativity λ2 0 Jn2 0 ... 0 ● Commutativity   ● Commutativity D = . . . ● Commutativity . . ● Eigenbases   ● Eigenbases  λk  ● Eigenbases  0 ... 0 Jnk  ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 36/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview The most celebrated and powerful result of linear algebra is More Linear Algebra now in reach. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Theorem 4 (The Jordan Normal Form) Given any matrix A, there is an ● Commutativity −1 ● Commutativity invertible matrix S such that A = SDS , where ● Rotation Matrices ● Rotation Matrices λ1 ● Commutativity Jn1 0 0 ... 0 ● Commutativity ● Commutativity λ2 0 Jn2 0 ... 0 ● Commutativity   ● Commutativity D = . . . ● Commutativity . . ● Eigenbases   ● Eigenbases  λk  ● Eigenbases  0 ... 0 Jnk  ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis I.e.: in the right basis, all matrices are of block-diagonal form, ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis where the blocks are sums of constant and standard nilpotent ● Behavioral Eigen-Analysis ● Nilpotency matrices. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 36/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview The most celebrated and powerful result of linear algebra is More Linear Algebra now in reach. ● Matrices Represent ALOT ● The Main Point ● A Practical Problem ● Commutativity Theorem 4 (The Jordan Normal Form) Given any matrix A, there is an ● Commutativity −1 ● Commutativity invertible matrix S such that A = SDS , where ● Rotation Matrices ● Rotation Matrices λ1 ● Commutativity Jn1 0 0 ... 0 ● Commutativity ● Commutativity λ2 0 Jn2 0 ... 0 ● Commutativity   ● Commutativity D = . . . ● Commutativity . . ● Eigenbases   ● Eigenbases  λk  ● Eigenbases  0 ... 0 Jnk  ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis   ● Behavioral Eigen-Analysis I.e.: in the right basis, all matrices are of block-diagonal form, ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis where the blocks are sums of constant and standard nilpotent ● Behavioral Eigen-Analysis ● Nilpotency matrices. ● Nilpotency D is the as-diagonalized-as-possible version of A. ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 36/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity   ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Restated, ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Restated, all vectors in a given J-block correspond to a single ● Behavioral Eigen-Analysis ● Nilpotency true eigenvector with eigenvalue λ ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Restated, all vectors in a given J-block correspond to a single ● Behavioral Eigen-Analysis ● Nilpotency true eigenvector with eigenvalue λ and multiplicity ni. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Restated, all vectors in a given J-block correspond to a single ● Behavioral Eigen-Analysis ● Nilpotency true eigenvector with eigenvalue λ and multiplicity ni. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form Problem 24 In the above terms, how many true eigenvectors does D have? ● A Simple Form ● A Simple Form ● A Simple Form - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together Putting it All Together

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity If is actually diagonalizable, block size , are ● Commutativity A ni = = 1 λi ● Commutativity actual eigenvalues; rows with same eigenvalues collect into ● Commutativity ● Commutativity eigenspaces. ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases In general, λi are generalized eigenvalues and like-valued ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis blocks collect into generalized eigenspaces. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis Restated, all vectors in a given J-block correspond to a single ● Behavioral Eigen-Analysis ● Nilpotency true eigenvector with eigenvalue λ and multiplicity ni. ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form Problem 24 In the above terms, how many true eigenvectors does D have? ● A Simple Form ● A Simple Form ● A Simple Form Answer: k = number of blocks. - p. 37/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity   ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Commutativity ● Eigenbases ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ■ Generalized eigenvectors get pushed “up" their J-block, ● Commutativity ● Commutativity asymptotically collapsing to a corresponding true ● Commutativity ● Eigenbases eigenvector. ● Eigenbases ● Eigenbases ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ■ Generalized eigenvectors get pushed “up" their J-block, ● Commutativity ● Commutativity asymptotically collapsing to a corresponding true ● Commutativity ● Eigenbases eigenvector. ● Eigenbases ● Eigenbases ■ Eigenvectors of real eigenvalues get stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ■ Generalized eigenvectors get pushed “up" their J-block, ● Commutativity ● Commutativity asymptotically collapsing to a corresponding true ● Commutativity ● Eigenbases eigenvector. ● Eigenbases ● Eigenbases ■ Eigenvectors of real eigenvalues get stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Behavioral Eigen-Analysis Eigenvectors of complex eigenvalues get rotated and ● Behavioral Eigen-Analysis stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ● Nilpotency ● Nilpotency ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ■ Generalized eigenvectors get pushed “up" their J-block, ● Commutativity ● Commutativity asymptotically collapsing to a corresponding true ● Commutativity ● Eigenbases eigenvector. ● Eigenbases ● Eigenbases ■ Eigenvectors of real eigenvalues get stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Behavioral Eigen-Analysis Eigenvectors of complex eigenvalues get rotated and ● Behavioral Eigen-Analysis stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Nilpotency Collapse, dilation, and rotation rates controlled by ● Nilpotency eigenvalues. ● Nilpotency ● Nilpotency ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together The Final Eigen-Analysis

● Overview J λ1 0 0 ... 0 More Linear Algebra n1 ● Matrices Represent ALOT 0 J λ2 0 ... 0 ● The Main Point  n2  ● A Practical Problem D = . . . ● Commutativity . . ● Commutativity   ● Commutativity  λk  ● Rotation Matrices  0 ... 0 Jnk  ● Rotation Matrices   ● Commutativity The final behavioral analysis:  ● Commutativity ● Commutativity ■ Generalized eigenvectors get pushed “up" their J-block, ● Commutativity ● Commutativity asymptotically collapsing to a corresponding true ● Commutativity ● Eigenbases eigenvector. ● Eigenbases ● Eigenbases ■ Eigenvectors of real eigenvalues get stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Behavioral Eigen-Analysis Eigenvectors of complex eigenvalues get rotated and ● Behavioral Eigen-Analysis stretched. ● Behavioral Eigen-Analysis ● Behavioral Eigen-Analysis ■ ● Nilpotency Collapse, dilation, and rotation rates controlled by ● Nilpotency eigenvalues. ● Nilpotency ● Nilpotency These are the only possible behaviors of a linear system. ● A Simple Form ● A Simple Form ● A Simple Form ● A Simple Form - p. 38/38 ● A Simple Form ● Putting it All Together ● Putting it All Together