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s11_mthsc853_hw10

# s11_mthsc853_hw10 - MthSc 853(Linear Algebra Dr Macauley HW...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 10 Due Monday, April 11th, 2011 (1) Let X be a finite-dimensional real Euclidean space. We say that a sequence { A n } of linear maps converges to a limit A if lim n →∞ || A n - A || = 0. (a) Show that { A n } converges to A if and only if for all x X , A n x converges to Ax . (b) Show by example that this fails if the dimension of X is infinite. (2) Let X be the space of continuous complex-valued functions on [ - 1 , 1] and define an inner product on X by ( f, g ) = Z 1 - 1 f ( s g ( s ) ds . Let m ( s ) be a continuous function of absolute value 1, that is, | m ( s ) | = 1, - 1 s 1. Define M to be multiplication by m : ( Mf )( s ) = m ( s ) f ( s ) . Show that M is unitary. (3) Let A be a linear map of a finite-dimensional complex Euclidean space X . (a) Show that A is normal if and only if it unitarily similar to a diagonal matrix, that is, A = U * DU for a diagonal matrix D and unitary matrix U . (b) Prove that if A is normal then it has a square-root, that is, a matrix B such that A = B 2 . Is B necessarily normal? Unique?
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