LetA={a}∈Rnxn, n IAl 1= max 1≤j≤n 2=1 llzll2= Vp(AT A)=maxtal,(A)} llalloo ∑ 三1 minim,nt lclr=】 a,P=VA4四=V空 =11=1 where (A)is the set of eigenvalues of A,p(A)=max,(A)}is called spectral radius. Copyright©2011，NAOYin Last Modification:Oct.2011
Let A = {aij} ∈ R n×n, kAk1 = max 1≤j≤n X n i=1 |aij| kxk2 = q ρ(ATA) = max{|λ|, λ ∈ σ(A)} kxk∞ = max 1≤i≤n X n j=1 |aij| kxkF = X n i=1 X n j=1 |aij| 2 = q tr(AAH) = vuut min X {m,n} i=1 σ 2 i where σ(A) is the set of eigenvalues of A, ρ(A) = max{|λ|, λ ∈ σ(A)} is called spectral radius. Copyright c 2011, NA Yin Last Modification: Oct. 2011 7
Compatible norms IAzl1≤IAlx1 ‖A2≤IAl2xll2 ‖Ax‖o≤‖lA‖llxl IAxl2≤‖A|Fx2 Another useful inequality between matrix norms is IAl2≤VIAlAIl Copyright©2011，NA⊙Yin Last Modification:Oct.2011 8
Compatible norms kAxk1 ≤ kAk1kxk1 kAxk2 ≤ kAk2kxk2 kAxk∞ ≤ kAk∞kxk∞ kAxk2 ≤ kAkF kxk2 Another useful inequality between matrix norms is kAk2 ≤ p kAk1kAk∞ Copyright c 2011, NA Yin Last Modification: Oct. 2011 8
Equivalence of norms LetA={a}∈Rmxn, lA2≤A‖F≤Vrllzlla2 ‖A‖F≤‖A*≤VTlxllE VA:≤4≤VmVF 1 VaAe≤Ae≤Vv where r rank(A) Copyright©2011，NAOYin Last Modification:Oct.2011 g
Equivalence of norms Let A = {aij} ∈ R m×n, kAk2 ≤ kAkF ≤ √ rkxk2 kAkF ≤ kAk∗ ≤ √ rkxkF 1 √ n kAk1 ≤ kAk2 ≤ √ m √ rkxk1 1 √ m kAk∞ ≤ kAk2 ≤ √ n √ rkxk∞ where r = rank(A). Copyright c 2011, NA Yin Last Modification: Oct. 2011 9
Permutation 2.0002 1.9998 1.9998 2.0002 2 =(④ Solution is x=(1,1)T. 0.0002 Let ob= -0.0002 2.0002 1.9998 4.0002 1.9998 2.0002 3.9998 Solution is=(1.5,0.5)T. lx-_1 lδblo-1 llalloo 2 bll∞ =2 Copyright©2011，NA⊙Yin Last Modification:Oct.2011 10
Permutation 2.0002 1.9998 1.9998 2.0002 x1 x2 = 4 4 Solution is x = (1, 1)T . Let δb = 0.0002 −0.0002 2.0002 1.9998 1.9998 2.0002 xe1 xe2 = 4.0002 3.9998 Solution is xe = (1.5, 0.5)T . kx − xek∞ kxk∞ = 1 2 kδbk∞ kbk∞ = 1 2 Copyright c 2011, NA Yin Last Modification: Oct. 2011 10