Eigenvalues and Eigenvectors - 5.1/ Determine if pairs 1,

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1 Eigenvalue and Eigenvector - 5./5.. Definition Definition An eigenvector of an n n matrix A i a nonzero vector x uch that Ax x for ome calar. A calar i called an eigenvalue of A if there i a nontrivial olution x of Ax x; uch an x i called an eigenvector correponding to. Note that the pair x uch that Ax x i called an eigenpair of A or a pair of eigenvalue and eigenvector of A. Example Let A. Determine if pair are pair of eigenvalue and eigenvector of A. Check if Ax x. 3 and 3. Computation of Eigenvalue and Their Correponding Eigenvector Obervation: a. Ax x Ax x A Ix. Since x i a nonzero vector the homogeneou ytem A Ix mut have nontrivial olution that i the matrix A I mut be ingular or deta I. Hence if x i an eigenpair A then i. i a olution of deta I ; ii. x i a nontrivial olution of A Ix. b. Let H x in R n ; A Ix. H i a ubpace of R n. H i called the eigenpace correponding to the eigenvalue. c. Let P deta I. P i a polynomial in with degree n. So the polynomial equation P ha the mot n real or complex olution. P i called the characteritic polynomial of A. Step for Computing x : a. Set the matrix A I. b. Compute P deta I; and olve P for. c. For each olved in b. find H x in R n ; A Ix Spanv v k where v v k i a bai for H. v v k are eigenpair of A correponding to eigenvalue. Obervation: a. If A i ingular then deta deta I i an eigenvalue of A. b. If A i a lower or upper triangular matrix then deta I a a a nn. So eigenvalue of A are diagonal element of A.

2 3 Example Find one eigenvalue and it correponding eigenpace of A A i ingular. So i an eigenvalue of A. Find H : 3 R R R 3 A 4 6 x x 3 x 3 x 3 R R 3 R 3. 3 t t H t 3 Span 3 Example Let A be a 7 7 matrix and I 7 be a 7 7 identify matrix. Suppoe we know deta I 7 5. Find all eigenvalue of A including it multiplicity. P deta I 7 5 Eigenvalue of A are: 5 i i 7 Example Find all eigenvalue of A where a. A 3 4 b. A a. A I 3 4 P deta I P 5 b. A I P deta I 4 3 P i 7

3 4 Example Find a bai for the eigenpace of A correponding to. Solve A Ix for x : A I 3 R R 3 R 3 3 x 3x x 3 x 3 x i free x a H a ; a real Span Example Find a bai for the eigenpace of A Solve A 3Ix for x : 3 A 3I correponding to R R R 3 R R 3 R 3 x x 3x 4 x x 3x 4 x 3t t t 3 H 3 t 3 ; t real Span 3 3. Propertie: a. A i ingular if and only if i an eigenvalue of A. Proof Let A be ingular. Then deta. Since deta I deta ian eigenvalue of A. Now let be an eigenvalue of A. Then deta I. Since deta deta I A i ingular. 3 b. If x i a pair of eigenvalue and eigenvector of A then k x i a pair of eigenvalue and eigenvector of A k. Proof Let x be a pair of eigenvalue and eigenvector of A. Then Ax x. A k x A k Ax A k x A k x... k Ax k x. Therefore k x i an eigenpair of A k.

4 c. Let A be noningular. If x i a pair of eigenvalue and eigenvector of A then x i a pair of eigenvalue and eigenvector of A. Proof Let x be a pair of eigenvalue and eigenvector of A. Then Ax x. Since and A exit Ax x A Ax A x A xva x x A x Hence x i a pair of eigenvalue and eigenvector of A. d. Eigenvalue of A and A T are the ame. Proof Oberve that deta T I deta T I T deta I. So eigenvalue of A and A T have the ame eigenvalue. e. Let x x x k be eigenvector of A correponding to ditinct eigenvalue k. Then the et of vector x x x k i linearly independent. Proof Suppoe that x x x k i linearly dependent. Then one of the vector i a linear combination of other vector. Aume that x i a linear combination of x x k and x x k i linearly independent (or having le number of vector). Let x x k x k. Then and x x k x k x k x k x Ax A x k x k Ax k Ax k x k k x k. Conider the difference of thee two equation: x k kx k. Since x x k i linearly independent the above equation ha only trivial olution i.e. k k k which contradict to the condition that k ditinct. So x cannot be a linear combination of x x k and x x x k i linearly independent. Example Let A be a 5 5 matrix with eigenvalue : 3 and 9. a. Determine if A exit and find all eigenvalue if it exit. b. Give all eigenvalue of A 3 T. a. Becaue none of the eigenvalue of A i A i noningular and A exit. Eigenvalue of A are: and Similarity 4

5 Definition Let A and B be n n matrice. A i aid to be imilar to B if there exit a noningular n n matrix P uch that P AP B or A PBP. Changing A to P AP i called a imilar tranformation. Propertie: a. A i imilar to itelf. I AI A. b. If A i imilar to B then B i imilar to A. P AP B PBP A c. If A i imilar to B and B i imilar to C then A i imilar to C. P AP B and P BP C P P AP P P P AP P P BP C. d. If A and B are imilar then A and B have the ame characteritic polynomial and therefore A and B have the ame eigenvalue. Let be an eigenvalue of A. Then P deta I. Since deta I deta I detp detp detp A IP detp AP I detb I be an eigenvalue of B. Example Let A be a 5 5 matrix with eigenvalue : 3 and 9. Find eigenvalue of B if If A and B are imilar 5. Ue TI-89 to Compute Eigenvalue and Eigenvector: MATH/Matrix (4)/eigV (9): eigenvalue MATH/Matrix (4)/eigVc (A): eigenvector Example: [;-3] STOA MATH/Matrix/eigv(A): MATH/Matrix/eigvc(A): So A ha eigenpair: