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prac3 - MAS 3114Practice Problem Set#3 1(a Dene what it...

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MAS 3114—Practice Problem Set #3 1. (a) Define what it means for two n × n matrices A and B to be similar. (b) Let A = bracketleftbigg 1 - 4 2 - 3 bracketrightbigg . Find a diagonal matrix D (with complex entries) such that A is similar to D . (c) Let A = bracketleftbigg 1 - 4 2 - 3 bracketrightbigg . Find a, b R such that A is similar to C = bracketleftbigg a - b b a bracketrightbigg . 2. Let A = bracketleftbigg 1 2 3 4 bracketrightbigg . (a) Carry out two iterations of the power method for calculating the dominant eigenvalue of A . Use vectore 1 as your starting vector. (b) Carry out two iterations of the inverse power method for calculating the eigenvalue of A which is closest to 0. Use vectore 1 as your starting vector. 3. Define a sequence of vectors vectorv k = bracketleftbigg x k y k bracketrightbigg by setting vectorv 0 = bracketleftbigg 1 0 bracketrightbigg and vectorv k +1 = Avectorv k for k 0, where A = bracketleftbigg 1 . 7 - . 3 - 1 . 2 . 8 bracketrightbigg .
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