Similarity If B=P-1AP where P is nonsingular,then A is similar with B,denoted by A B. ●lfA~B,thenλ(A)=入(B);if z is a eigenvector of B,then Pz is that of A. Copyright©2011，NA⊙Yin Last Modification::Oct.2011 7
Similarity If B = P −1AP where P is nonsingular, then A is similar with B, denoted by A ∼ B. • If A ∼ B, then λ(A) = λ(B); if z is a eigenvector of B, then P z is that of A. Copyright c 2011, NA Yin Last Modification: Oct. 2011 7
Reilaigh Quotient If A is s.p.d.,and Ai,(i=1,2,...,n)are the eigenvalues of A such that 入1≥2≥..≥入n,then X1≥，4@≥x （x,c) (x,Ax) 入1=max x≠0（x,x) 入n=mih (z,Az) c≠0(x,r) Copyright©2011，NA⊙Yin Last Modification:Oct.2011 8
Reilaigh Quotient If A is s.p.d., and λi ,(i = 1, 2, . . . , n) are the eigenvalues of A such that λ1 ≥ λ2 ≥ . . . ≥ λn, then λ1 ≥ (x, Ax) (x, x) ≥ λn λ1 = max x6=0 (x, Ax) (x, x) λn = min x6=0 (x, Ax) (x, x) Copyright c 2011, NA Yin Last Modification: Oct. 2011 8