Diagonalizability Norms Lecture 3 Introduction Amos Ron University of Wisconsin Madison January 29 2021 Amos Ron CS513 remote learning S21 Diagonalizability Norms Outline 1 Diagonalizability General square case The Symmetric case the Schur decomposition 2 Norms Vector Norms Matrix Norms Amos Ron CS513 remote learning S21 Diagonalizability Norms General square case The Symmetric case the Schur decomposition Outline 1 Diagonalizability General square case The Symmetric case the Schur decomposition 2 Norms Vector Norms Matrix Norms Amos Ron CS513 remote learning S21 Diagonalizability Norms General square case The Symmetric case the Schur decomposition Blank page Amos Ron CS513 remote learning S21 Diagonalizability Norms General square case The Symmetric case the Schur decomposition 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 Diagonalizability Norms General square case The Symmetric case the Schur decomposition 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 Diagonalizability Norms General square case The Symmetric case the Schur decomposition 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 Diagonalizability Norms General square case The Symmetric case the Schur decomposition 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 Diagonalizability Norms General square case The Symmetric case the Schur decomposition The symmetric case Reminder A is symmetric whenever A A cid 48 Theorem Spectral rudiments of a symmetric matrix Assume A A cid 48 Then A R A is diagonalizable There is an A eigenbasis which is also an orthonormal basis The Schur Decomposition A is orthogonally diagonalizable A QDQ cid 48 QDQ 1 with Q orthogonal and D diagonal Amos Ron CS513 remote learning S21 Diagonalizability Norms General square case The Symmetric case the Schur decomposition Demo 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms Outline 1 Diagonalizability General square case The Symmetric case the Schur decomposition 2 Norms Vector Norms Matrix Norms Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of Norm De nition Norm Let be an assignment from Rm to R c R c 0 Rm cid 51 v cid 55 v R This assignment is a norm if the following conditions are valid v 0 if and only if v 0 For c R v Rm we have cv c v For v w Rm v w v w Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of Norm Example The 1 norm mean norm cid 96 1 norm v 1 v i m cid 88 i 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of Norm Example The 2 norm Euclidean norm cid 96 2 norm the least square norm v 2 v i 2 cid 118 cid 117 cid 117 cid 116 m cid 88 i 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of Norm Example The norm max norm cid 96 norm uniform norm v max 1 i m v i Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of Norm Example The p norm cid 96 p norm 1 p v p cid 33 1 p v i p cid 32 m cid 88 i 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of matrix norms A is m n maps thus Rn to Rm We choose a norm for the domain and a norm cid 48 for the range Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of matrix norms A is m n maps thus Rn to Rm We choose a norm for the domain and a norm cid 48 for the range De nition Matrix norm Av cid 48 v A max v cid 54 0 max Av cid 48 v 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms De nition of matrix norms De nition Matrix norm Av cid 48 v A max v cid 54 0 max Av cid 48 v 1 If the norms and cid 48 are both p norms for the same p we denote the matrix norm as A p Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms The 1 norm of a matrix Theorem computing A 1 Let Am n with columns a1 an Then A 1 max 1 i n ai 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms The 1 norm of a matrix Theorem computing A 1 Let Am n with columns a1 an Then A 1 max 1 i n ai 1 X Proof We need to show that A 1 X and A 1 X First for any 1 j m ej 1 1 therefore aj 1 Aej 1 A 1 Therefore X A 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms The 1 norm of a matrix Theorem computing A 1 Let Am n with columns a1 an Then A 1 max 1 i n ai 1 X Now let v Rn v 1 Then Av 1 v i ai 1 v i ai 1 v i ai 1 n cid 88 i 1 n cid 88 i 1 n cid 88 i 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms The 1 norm of a matrix Now let v Rn v 1 Then Av 1 v i ai 1 v i ai 1 v i ai 1 n cid 88 i 1 n cid 88 i 1 n cid 88 i 1 v i X n cid 88 i 1 Amos Ron CS513 remote learning S21 Diagonalizability Norms Vector Norms Matrix Norms The 1 norm of a matrix Now let v Rn v 1 Then Av 1 v i ai 1 v i ai 1 v i ai 1 n cid 88 i 1 n cid 88 i 1 n cid 88 i 1 v i X n cid 88 i 1 X v i X v …