hw4 - = • 3 2 1 2 ‚ q = • 1 2 ‚ 4 Let X ∈ R n ×...

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Assignment 4 Math 570A Fall 2009 1) Let A,B C n × n . Recall λ ( A ) is the set of eigenvalues of A . Show that λ ( AB ) = λ ( BA ). 2) For the matrix A = 1 - 3 - 2 2 find a matrix X such that X - 1 AX is diagonal with the eigenvalues of A on the diagonal (see example 7.1.4) 3) Do 3 iterations of the power method, one iteration of the Rayleigh quotient method, and one iteration of the QR method for finding eigenvalues using the matrix and initial vector A
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Unformatted text preview: = • 3 2 1 2 ‚ q = • 1 2 ‚ 4) Let X ∈ R n × n be nonsingular. (a) Show that k A k X = k X-1 AX k p is a matrix norm. (b) Show that k AB k X ≤ k A k X k B k X . (c) Show that for a diagonal matrix D ∈ R n × n we have that k D k p = max 1 ≤ i ≤ n | d ii | (d) For a nondegenerate A ∈ R n × n find a nonsingular X such that k A k X = max λ ∈ λ ( A ) | λ | 1...
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