s11_mthsc853_hw11

# s11_mthsc853_hw11 - j ), and show that there is an...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 11 Due Wednesday, April 20th, 2011 Throughout, X is a ﬁnite-dimensional Euclidean space. (1) Deﬁne the index of a real symmetric matrix A to be the number of strictly positive eigenvalues minus the number of strictly negative eigenvalues. Suppose A and B are real symmetric matrices and x T Ax x T Bx for all x X . Prove that the index of A is at most the index of B . (2) Let H,M : X X be self-adjoint mappings, and M positive deﬁnite. Deﬁne R H,M ( x ) = ( x,Hx ) ( x,Mx ) . (a) Let μ = inf { R H,M ( x ) | x X } . Show that μ exists, and that there is some v X for which R H,M ( v ) = μ , and that μ and v satisfy Hv = μMv . (b) Show that the constrained minimum problem min { R H,M ( y ) | ( y,Mv ) = 0 } has a nonzero solution w X , and that this solution satisﬁes Hw = κMw , where κ = R H,M ( w ). (3) Let H,M : X X be self-adjoint mappings, and M positive deﬁnite. (a) Show that there exists a basis v 1 ,...,v n of X where each v i satisﬁes an equation of the form Hv i = μ i Mv i ( μ i real) , ( v i ,Mv j ) = ± 1 i = j 0 i 6 = j (b) Compute ( v i ,Hv
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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 self-adjoint mappings, and M positive denite. (a) Prove that all the eigenvalues of M-1 H are real. (b) Prove that if H is positive-denite, then all the eigenvalues of M-1 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 matrix-valued function that is dierentiable and invertible. Use the product rule ( d dt [ A ( t ) B ( t )] = AB + A B ) to derive d dt A-1 =-A-1 d dt A A-1 ....
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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.

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