assign4b - A and B are similar n n matrices, then tr( A ) =...

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MATH 235/W08 Assignment 4B Eigenvalues, Eigenvectors, Diagonalization Do NOT hand in this assignment. 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 , respectively. Consider the vector v = x + y + z . Can v be an eigenvector of A corresponding to an eigenvalue λ (possibly diFerent 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( A ) = n i =1 a ii . (a) Let C and D be any two m × n matrices. Prove that tr( C T D ) = tr( DC T ). (b) Prove that if
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Unformatted text preview: A and B are similar n n matrices, then tr( A ) = tr( B ) and det( A ) = det( 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. Let A and B be invertible n n matrices. Prove that AB and BA have the same eigenvalues. 5. Determine if the following pair of matrices are similar to each other: A = 1 1 5 2 2 , B = 1 7 2 7 2 . 1...
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