Announcements Monday, November 5 I The third midterm is on Friday, November 16. I That is one week from this Friday. I The exam covers xx4.5, 5.1, 5.2. 5.3, 6.1, 6.2, 6.4, 6.5. I WeBWorK 6.1, 6.2 are due Wednesday at 11:59pm. I The quiz on Friday covers xx6.1, 6.2. I My office is Skiles 244 and Rabinoffice hours are: Mondays, 12{1pm; Wednesdays, 1{3pm. Section 6.4 Diagonalization Motivation Difference equations Many real-word linear algebra problems have the form: 2 3 n v1 = Av0; v2 = Av1 = A v0; v3 = Av2 = A v0;::: vn = Avn−1 = A v0: This is called a difference equation. Our toy example about rabbit populations had this form. The question is, what happens to vn as n ! 1? I Taking powers of diagonal matrices is easy! I Taking powers of diagonalizable matrices is still easy! I Diagonalizing a matrix is an eigenvalue problem. 2 0 4 0 8 0 2n 0 D = ; D2 = ; D3 = ;::: Dn = : 0 −1 0 1 0 −1 0 (−1)n 0 −1 0 0 1 0 1 0 0 1 0 −1 0 0 1 1 2 1 3 1 D = @ 0 2 0 A ; D = @ 0 4 0 A ; D = @ 0 8 0 A ; 1 1 1 0 0 3 0 0 9 0 0 27 0 (−1)n 0 0 1 n 1 ::: D = @ 0 2n 0 A 1 0 0 3n Powers of Diagonal Matrices If D is diagonal, then Dn is also diagonal; its diagonal entries are the nth powers of the diagonal entries of D: (CDC −1)(CDC −1) = CD(C −1C)DC −1 = CDIDC −1 = CD2C −1 (CDC −1)(CD2C −1) = CD(C −1C)D2C −1 = CDID2C −1 = CD3C −1 CDnC −1 Closed formula in terms of n: easy to compute 1 1 2n 0 1 −1 −1 1 2n + (−1)n 2n + (−1)n+1 = : 1 −1 0 (−1)n −2 −1 1 2 2n + (−1)n+1 2n + (−1)n Powers of Matrices that are Similar to Diagonal Ones What if A is not diagonal? Example 1=2 3=2 Let A = . Compute An, using 3=2 1=2 1 1 2 0 A = CDC −1 for C = and D = : 1 −1 0 −1 We compute: A2 = A3 = . An = Therefore An = A = CBC −1; B = EDE −1 =) A = C(EDE −1)C −1 = (CE)D(CE)−1: Similar Matrices Definition Two n × n matrices are similar if there exists an invertible n × n matrix C such that A = CBC −1. Fact: if two matrices are similar then so are their powers: A = CBC −1 =) An = CBnC −1: Fact: if A is similar to B and B is similar to D, then A is similar to D. Diagonalizable Matrices Definition An n × n matrix A is diagonalizable if it is similar to a diagonal matrix: A = CDC −1 for D diagonal. Important 0 d11 0 ··· 0 1 B 0 d22 ··· 0 C If A = CDC −1 for D = B C then B . .. C @ . A 0 0 ··· dnn 0 k 1 d11 0 ··· 0 k B 0 d22 ··· 0 C Ak = CDK C −1 = C B C C −1: B . .. C @ . A k 0 0 ··· dnn So diagonalizable matrices are easy to raise to any power. Diagonalization The Diagonalization Theorem An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In this case, A = CDC −1 for 0 λ1 0 ··· 0 1 0 j j j 1 B 0 λ2 ··· 0 C C = v v ··· v D = B C ; @ 1 2 n A B . .. C j j j @ . A 0 0 ··· λn where v1; v2;:::; vn are linearly independent eigenvectors, and λ1; λ2; : : : ; λn are the corresponding eigenvalues (in the same order). Corollary a theorem that follows easily from another theorem An n × n matrix with n distinct eigenvalues is diagonalizable. The Corollary is true because eigenvectors with distinct eigenvalues are always linearly independent. We will see later that a diagonalizable matrix need not have n distinct eigenvalues though. Diagonalization The Diagonalization Theorem An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In this case, A = CDC −1 for 0 λ1 0 ··· 0 1 0 j j j 1 B 0 λ2 ··· 0 C C = v v ··· v D = B C ; @ 1 2 n A B . .. C j j j @ . A 0 0 ··· λn where v1; v2;:::; vn are linearly independent eigenvectors, and λ1; λ2; : : : ; λn are the corresponding eigenvalues (in the same order). Note that the decomposition is not unique: you can reorder the eigenvalues and eigenvectors. 0 j j 1 0 j j 1−1 0 j j 1 0 j j 1−1 λ 0 λ 0 A = v v 1 v v = v v 2 v v @ 1 2 A 0 λ @ 1 2 A @ 2 1 A 0 λ @ 2 1 A j j 2 j j j j 1 j j This is a triangular matrix, so the eigenvalues are the diagonal entries 1; 2; 3. A diagonal matrix just scales the coordinates by the diagonal entries, so we can take our eigenvectors to be the unit coordinate vectors e1; e2; e3. Hence the Diagonalization Theorem says 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 @ 0 2 0 A = @ 0 1 0 A @ 0 2 0 A @ 0 1 0 A : 0 0 3 0 0 1 0 0 3 0 0 1 It doesn't give us anything new because the matrix was already diagonal! Diagonalization Easy example Question: What does the Diagonalization Theorem say about the matrix 0 1 0 0 1 A = @ 0 2 0 A? 0 0 3 A diagonal matrix D is diagonalizable! It is similar to itself: −1 D = InDIn : The characteristic polynomial is f (λ) = λ2 − Tr(A) λ + det(A) = λ2 − λ − 2 = (λ + 1)(λ − 2): Therefore the eigenvalues are −1 and 2. Let's compute some eigenvectors: 3=2 3=2 rref 1 1 (A + 1I )x = 0 () x = 0 x = 0 3=2 3=2 0 0 −1 The parametric form is x = −y, so v1 = 1 is an eigenvector with eigenvalue −1. −3=2 3=2 rref 1 −1 (A − 2I )x = 0 () x = 0 x = 0 3=2 −3=2 0 0 1 The parametric form is x = y, so v2 = 1 is an eigenvector with eigenvalue 2. The eigenvectors v1; v2 are linearly independent, so the Diagonalization Theorem says −1 1 −1 0 A = CDC −1 for C = D = : 1 1 0 2 Diagonalization Example 1=2 3=2 Problem: Diagonalize A = . 3=2 1=2 The characteristic polynomial is f (λ) = det(A − λI ) = −λ3 + 4λ2 − 5λ + 2 = −(λ − 1)2(λ − 2): Therefore the eigenvalues are 1 and 2, with respective multiplicities 2 and 1. Let's compute the 1-eigenspace: 0 1 0 1 3 −3 0 rref 1 −1 0 (A − I )x = 0 () @ 2 −2 0 A x = 0 @ 0 0 0 A x = 0 1 −1 0 0 0 0 The parametric vector form is x = y 0x1 011 001 y = y =) @yA = y @1A + z @0A z = z z 0 1 Hence a basis for the 1-eigenspace is 011 001 B1 = v1; v2 where v1 = @1A ; v2 = @0A : 0 1 Diagonalization Another example 0 4 −3 0 1 Problem: Diagonalize A = @ 2 −1 0 A. 1 −1 1 Now let's compute the 2-eigenspace: 0 1 0 1 2 −3 0 rref 1 0 −3 (A − 2I )x = 0 () @ 2 −3 0 A x = 0 @ 0 1 −2 A x = 0 1 −1 −1 0 0 0 The parametric form is x = 3z; y = 2z, so an eigenvector with eigenvalue 2 is 031 v3 = @2A : 1 The eigenvectors v1; v2; v3 are linearly independent: v1; v2 form a basis for the 1-eigenspace, and v3 is not contained in the 1-eigenspace. Therefore the Diagonalization Theorem says 0 1 0 3 1 0 1 0 0 1 −1 A = CDC for C = @ 1 0 2 A D = @ 0 1 0 A : 0 1 1 0 0 2 Diagonalization Another example, continued 0 4 −3 0 1 Problem: Diagonalize A = @ 2 −1 0 A. 1 −1 1 Note: In this case, there are three linearly independent eigenvectors, but only two distinct eigenvalues. This is an upper-triangular matrix, so the only eigenvalue is 1. Let's compute the 1-eigenspace: 0 1 (A − I )x = 0 () x = 0: 0 0 This is row reduced, but has only one free variable x; a basis for the 1 1 1-eigenspace is f 0 g. So all eigenvectors of A are multiples of 0 . Diagonalization A non-diagonalizable matrix 1 1 Problem: Show that A = is not diagonalizable. 0 1 Conclusion: A has only one linearly independent eigenvector, so by the \only if" part of the diagonalization theorem, A is not diagonalizable. Poll Which of the following matrices are diagonalizable, and why? 1 2 1 2 2 1 2 0 A. B. C. D. 0 1 0 2 0 2 0 2 MatrixA is not diagonalizable: its only eigenvalue is 1, and its 1-eigenspace is 1 spanned by 0 . Similarly, matrixC is not diagonalizable. MatrixB is diagonalizable because it is a 2 × 2 matrix with distinct eigenvalues. MatrixD is already diagonal! Poll Diagonalization Procedure How to diagonalize a matrix A: 1. Find the eigenvalues of A using the characteristic polynomial.