M235A4a - MATH 235 Diagonalization - 1 Assignment 4a Not to...

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MATH 235 Diagonalization - 1 Assignment 4a Not to be handed in. 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 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( 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
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