Unformatted text preview: M such that A = M1 ¯ AM Obtain the ±ollowing ±or two similar matrices A and ¯ A . (i) Show that A m is similar to ¯ A m ±or any m ≥ 1, where the superscript “ m ” denotes matrix product, A 1 = A, A m = A ( A m1 ) , m ≥ 1 . (ii) Show that v is an eigenvector ±or A i± and only i± Mv is an eigenvector ±or ¯ A . (iii) Suppose that ¯ A is diagonal ( ¯ A ij = 0 i± i n = j ). Suppose moreover that  ¯ A ii  < 1 ±or each i . Conclude that I − A admits an inverse by combining Prob. 2 and Prob. 3 (i). 4 . Which o± the ±ollowing sets are linearly independent? (a) 1 3 2 , 1 , 1 in ( R 3 , R ). (b) 1 3 2 , 1 , 1 , 1 √ 2 π , in ( R 3 , R ). (c) 1 3 2 , 1 , 1 , 1 √ 2 j π , in ( C 3 , C )....
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 Spring '08
 Ma
 Linear Algebra, Matrices, Invertible matrix

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