HW3_602_2012 - Math 602(2012 Homework 3 1 Let A be a...

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Unformatted text preview: Math 602 (2012) Homework 3 1. Let A be a self-adjoint matrix. Show that there is a self-adjoint matrix B so that B 3 = A(A - 2I)(A - 3I) What are the eigenvalues of B in terms of the eigenvalues of A? What are the eigenvectors of B in terms of the eigenvectors of A? Will B commute with A? 2. Assume that A is a symmetric matrix with eigenvalues not equal to 3. True or false? The matrices A(A - 2I) and (A - I)(A - 3I)-1 commute. 3. Show that if N is a normal matrix then N x = N x for all vectors x. And from Strang, 3rd Ed. (Caution! this is the old edition, see the problems on Carmen), p.319 solve: 5.3, 5.4, 5.5, 5.19, 5.20(a)(b). 1 ...
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