assig4M235W08 - A = ∑ n i =1 a ii(a Let C and D be any...

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Math 235 Linear Algebra II Assignment 4 Due 9:30am, Wednesday Oct 8, 2008, in the drop box designated for your section. 1. Let A be an n × n matrix. Also, let α,β,γ be three distinct eigenvalues of A having corresponding eigenvectors x , y , z . Consider the vector v = x + y + z . Can v be an eigenvector of A corresponding to an eigenvalue λ (possibly different from α,β,γ )? Explain. 2. Recall that the trace of an n × n matrix A = [ a ij ], denoted by tr( A ), is the sum of the diagonal elements, that is, tr(
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Unformatted text preview: A ) = ∑ n i =1 a ii . (a) Let C and D be any two n × n matrices. Prove that tr( CD ) =tr( DC ). (b) Prove that if A and B are similar n × n matrices, then tr( A ) =tr( B ). 3. Let M , N be n × n matrices and suppose that M is invertible. Denote A = MN and B = NM . Prove that A is similar to B . 4. From the Text § 5.1 # 2, 6, 16, 22, 25, 26, 27, 32 § 5.2 # 14, 16, 20, 24...
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