Matrix Norms Positive de nite matrices Lecture 4 Introduction Amos Ron University of Wisconsin Madison February 01 2021 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Outline 1 Matrix Norms Characterizing the norm Characterizing the 2 norm 2 Positive de nite matrices De nition and example Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Outline 1 Matrix Norms Characterizing the norm Characterizing the 2 norm 2 Positive de nite matrices De nition and example Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Blank page Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm norm Theorem Computing the norm 1 For an A m n A 1 A cid 48 2 Let b cid 48 1 b cid 48 m be the rows of A Then A max 1 i m bi 1 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm norm Theorem Computing the norm 1 For an A m n A 1 A cid 48 2 Let b cid 48 1 b cid 48 m be the rows of A Then A max 1 i m bi 1 Comment The equivalence of the two conditions above follows directly from the characterization of the 1 norm Comment Assertion 2 above can be proved directly using a similar approach but with different details to the prooof of the 1 norm case Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm norm We show how to prove A 1 A cid 48 directly from basic Linear Algebra principles Step I Show that for any v Rm v 1 max v w w 1 and v max v w w 1 1 Step II Since A 1 max Av 1 v 1 1 it follows that A 1 max Av w v 1 1 w 1 Step III Since A cid 48 max A cid 48 w w 1 it follows that A cid 48 max A cid 48 w v v 1 1 w 1 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm The 2 norm of a matrix Some basics v 2 2 v v Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm The 2 norm of a matrix Some basics v 2 2 v v Whatever A is A cid 48 A is symmetric and its eigenvalues are non negative BC cid 48 C cid 48 B cid 48 A cid 48 A cid 48 A cid 48 A cid 48 cid 48 A cid 48 A Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm The 2 norm of a matrix Some basics v 2 2 v v Whatever A is A cid 48 A is symmetric and its eigenvalues are non negative A cid 48 A v v v 2 2 v v A cid 48 Av v Av Av Av 2 2 Also 0 Av 2 2 v 2 2 A 2 2 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm The 2 norm of a matrix Whatever A is A cid 48 A is symmetric and its eigenvalues are non negative De nition A right singular vector of A is an eigenvector of A cid 48 A An s 0 is a singular value of A is s2 A cid 48 A Notation spectral radius A square A max A So A 2 cid 112 A cid 48 A Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Characterizing the 2 norm Theorem Chracterizing the 2 norm A 2 cid 112 A cid 48 A i e A 2 the largest singular value of A Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Characterizing the 2 norm Theorem Chracterizing the 2 norm A 2 cid 112 A cid 48 A i e A 2 the largest singular value of A Proof We already saw that A 2 cid 112 A cid 48 A Now Let v Rm such that v 2 1 and A 2 Av 2 Let A cid 48 A QDQ cid 48 be the Schur decomposition of A cid 48 A Then A 2 2 Av 2 2 Av Av A cid 48 Av v QDQ cid 48 v v DQ cid 48 v Q cid 48 v Denote w Q cid 48 v Since Q cid 48 is orthogonal w 2 v 2 1 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Characterizing the 2 norm A 2 2 Av 2 2 Av Av A cid 48 Av v QDQ cid 48 v v DQ cid 48 v Q cid 48 v Denote w Q cid 48 v Since Q cid 48 is orthogonal w 2 v 2 1 So A 2 2 Dw w D i i w i 2 A cid 48 A w i 2 A cid 48 A w i 2 A cid 48 A m cid 88 i 1 m cid 88 i 1 m cid 88 i 1 So A 2 cid 112 A cid 48 A Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices Characterizing the norm Characterizing the 2 norm Demo 2 Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices De nition and example Outline 1 Matrix Norms Characterizing the norm Characterizing the 2 norm 2 Positive de nite matrices De nition and example Amos Ron CS513 remote learning S21 Matrix Norms Positive de nite matrices De nition and example De nition of Positive De niteness Amos Ron CS513 remote learning S21