Final Exam Review 1 Selected Answers Math 4242 Summer 2018

1. Write a clear and concise definition for the following terms. • Elementary row operations and • Orthonormal basis corresponding matrices • Gram-Schmidt process • Gaussian elimination • Orthogonal matrix (proper and improper) • Regular matrix (as it relates to Gaussian • Orthogonal projection of a vector eliminiation) • Orthogonal subspaces • Pivots • Orthogonal complement • Nonsingular matrix • Linear function/operator/functional • Matrix inverse • Linear transformation • Gauss-Jordan elimination • Dual space • Similar matrices • Rotation, reflection, stretch, shear matrices • Symmetric matrix • Change of basis formula • Homogeneous/Nonhomogeneous linear system • Linear system • Rank of a matrix • Superposition principle for general linear systems • Free variable • Range, corange, kernel, cokernel of a general • Determinant linear function • Trace • Adjoint of a linear function • Vector space • Eigenvalue and eigenvector • Subspace • Characteristic polynomial and equation • Linear combination • Algebraic and geometric multiplicity of • Span of a set of vectors an eigenvalue • Linear dependence and independence • Eigenspace • Basis • Complete matrix • Dimension • Minimizer of a quadratic form • Expansion of a vector in a basis • Normal equations, weighted normal equations • Superposition Principle • Least squares solution • Range, Corange, Kernel, Cokernel • Probability vector • Adjoint system • Transition matrix • Inner product • Markov chain • Orthogonal vectors • Regular matrix (as it relates to Markov chains), • Norm regular Markov chain • Unit vector • Positive definite, positive semi-definite • Quadratic form • Gram matrix 2. Describe the following matrix factorizations. Write down what types of matrices you can use the facotrizations on and what types of matrices are in the facorization. (a) LU → §1.3 (b) LDV → §1.5 (c) LDLT → §1.6 (d) QR facorization → §4.3 (e) SΛS−1 (diagonalization) → §8.3 (f) QΛQT (spectral factorization) → §8.5 (g) SJS−1 (Jordan decomposition) → §8.6 3. Find the eigenvalues and eigenvectors of the following matrices. What are the algebraic and geometric multiplicities of each of the eigenvalues? Find the Jordan canonical form for each matrix. How many Jordan blocks does each Jordan matrix have? You need not explicitly find any inverses here. 1 1 0   (a) A = 0 1 1 0 0 4 Eigenvalues:

λ1 = 4, λ2 = 1 Eigenvectors: 1 1     v1 = 3 , v2 = 0 9 0 Multiplicity:

λ1 has algebraic multiplicity 1 and geometric multiplicity 1

λ2 has algebraic multiplicity 2 and geometric multiplicity 1 Jordan Canonical Form: There are two Jordan blocks in the Jordan canonical form J. We see this from the fact that the sum of the geometric multiplicities is 1+1 = 2. 1 1 0   J = 0 1 0 0 0 4 We ﬁnd the Jordan canonical form by putting the eigenvalues on the diagonal, and the 1’s on the superdiagonal as needed to separate our blocks.   2 9 0 2   −1 2 1 0  (b) A =    0 0 3 0    0 0 1 −1 Eigenvalues:

λ1 = 2 + 3i, λ2 = 2 − 3i, λ3 = 3, λ4 = −1 Eigenvectors:

        −3i 3i 19 −3          1   1   1  −1 v1 =   , v2 =   , v3 =   , v4 =    0   0  20  0          0 0 5 9

Multiplicity: All four eigenvalues have algebraic and geometric multiplicity 1. Jordan Canonical Form (Diagonalizable): There are four Jordan blocks in the Jordan canonical form J.

  −1 0 0 0    0 2 − 3i 0 0 J =    0 0 2 + 3i 0   0 0 0 3

4. Find the angle between x and x2 − 1 on the interval [0, 1] with respect to the L2 inner product. Angle between two functions f and g is found by using the formula

hf, gi cos(θ) = kfk kgk

where the inner product and norm used in this case is the L2 inner product and norm. So we compute what we need and then put it into the formula to solve.

Z 1 2 2 1 hx, x − 1iL2 = = x(x − 1) dx = − 0 4 1 Z 1 2 1 kxkL2 = x · x dx = √ 0 3 1 Z 1 2 r 2 2 2 8 kx − 1kL2 = (x − 1) · (x − 1) dx = 0 15 1 r r ! 4 3 5 3 5 cos(θ) = q = · =⇒ θ = arccos · ≈ 0.9361 rad √1 · 8 8 2 8 2 3 15 5. Find the minimizer and minimum of the quadratic form

q(x) = x2 + 3xy + 3y2 − 2xz + 8z2.

First we need to put this in its matrix form q(x) = xT Kx − 2xT f + c:

 3    1 2 −1 0 K =  3  f =   c = 0  2 3 0  0 −1 0 8 0 So we need to solve the normal equations, Kx = f (given that f is the zero vector, if K is positive definite, then I think the minimizer is probably the zero vector). If K is positive definite, it is a unique minimizer. You can show that this is regular and that the pivots are all positive so it is in fact positive definite. Then the minimum and minimizer are:

q(0, 0, 0) = 0.

6. Find a basis for the four fundamental subspaces of the matrix 5 0   A = 1 2 . 0 2 Using row reduction, we can see that     5 0 1 0 ! ! 5 1 0 1 0 − 1 A = 1 2 → 0 1 AT = → 5     0 2 2 0 1 1 0 2 0 0 So now we want to ﬁnd the bases for the four fundamental subspaces.

(1) Kernel:   1 0 0 ( !) 0 0 1 0 =⇒ ker(A) = span   0 0 0 0 (2) Image: Pivot is ﬁrst two columns, thus 5 0       img(A) = span 1 , 2    0 2  (3) Cokernel:   !  1 1 0 − 1 0 x − 1 t = 0   5 5   =⇒ =⇒ coker(A) = span −5 0 1 1 0 y + t = 0    5 

Another way we could ﬁnd the basis vector for the cokernel is to notice that coker(A) ⊥ img(A). Since the image is two dimensional, we know that the cokernel, being orthogonal, must be 1 dimen- T sional. If v = (v1, v2, v3) was in the cokernel, then we know that  5v1 + v2 = 0

2v2 + v3 = 0 Solving these equations, we get the same answer:  1      coker(A) = span −5    5  (4) Coimage: Pivot is ﬁrst two columns, thus ( ! !) 5 1 coimg(A) = span , 0 2 7. Write out all the ways that you know how to determine if a matrix is singular or nonsingular.