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Unformatted text preview: j ), and show that there is an invertible real matrix U for which U * MU = I and U * HU is diagonal. (c) Characterize the numbers 1 ,... n by a minimax principle. (4) Let H,M : X X be selfadjoint mappings, and M positive denite. (a) Prove that all the eigenvalues of M1 H are real. (b) Prove that if H is positivedenite, then all the eigenvalues of M1 H are positive. (c) Show by example that (a) and (b) fail if M is not positive denite. (5) Let N : X X be a normal mapping of a Euclidean space. Prove that  N  = max  n i  , where the n i s are the eigenvalues of N . (6) Let A ( t ) be a matrixvalued function that is dierentiable and invertible. Use the product rule ( d dt [ A ( t ) B ( t )] = AB + A B ) to derive d dt A1 =A1 d dt A A1 ....
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This note was uploaded on 03/11/2012 for the course MTHSC 853 taught by Professor Staff during the Spring '08 term at Clemson.
 Spring '08
 Staff
 Linear Algebra, Algebra

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