e vectors e values Lecture 2 Introduction Amos Ron University of Wisconsin Madison January 27 2021 Amos Ron CS513 remote learning S21 e vectors e values Outline 1 e vectors e values De nition example Diagonalizability Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Outline 1 e vectors e values De nition example Diagonalizability Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs De nition A m m v C v Cm 0 is eigenpair of A if The set of all eigenvalues is the spectrum Av v A of A Note A real valued matrix might have complex eigenvalues Reminder The characteristic polynomial of A is pA t det A tI A pA 0 Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs The eigenvectors of the matrix A is any linearly independent maximal set of eigenvectors The cardinality of such set depends only on A Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs A cid 19 cid 18 2 2 1 3 Then pA t t 2 t 3 2 t2 5t 4 This implies that A 1 4 Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs Then The matrix A 4I must be singular Indeed A cid 19 cid 18 2 2 1 3 A 4I cid 19 cid 18 2 2 1 1 Every non zero vector in ker A 4I is an eigenvector Since dim ker A 4I 1 we select only one eigenvector from this null space For example we can choose v1 Then 4 v1 is an eigenpair Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs Then The matrix A I must be singular Indeed A cid 19 cid 18 2 2 1 3 A I cid 19 cid 18 1 2 1 2 Every non zero vector in ker A I is an eigenctor Since dim ker A I 1 we select only one eigenvector from this null space For example v2 Then 1 v2 is an eigenpair Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Eigenpairs A cid 19 cid 18 2 2 1 3 Then Since eigenvectors associated with different e values are always linearly independent v1 v2 are independent hence form a basis for R2 Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Diagonalizability De nition Diagonalizability A m m is diagonalizable is there exists a basis for Cm made of e vectors of A Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Diagonalizability Theorem A is square TFCAE 1 A is diagonalizable 2 There exist a matrix P and a diagonal matrix D such that A PDP 1 3 Another equivalent condition deferred Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Proof of Theorem 2 1 We have AP PD We prove that each column of P is an eigenvector of A This proves 1 since the columns of any m m invertible matrix form a basis for Cm The jth column of P is Pej Now A Pej AP ej PD ej P Dej P D j j ej D j j Pej So D j j Pej is an eigenpair of A Amos Ron CS513 remote learning S21 e vectors e values De nition example Diagonalizability Proof of Theorem 1 2 We are given m eigenpairs j vj with v1 vm a basis for Cm Let P be the matrix whose columns are v1 vj and let D be the diagonal matrix whose diagonal is 1 m We show that A PDP 1 by showing that AP PD i e by showing that for every j AP ej PD ej Now AP ej A Pej Avj jvj P jej P Dej PD ej Amos Ron CS513 remote learning S21