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Exam_4exam_ _ - MATH 405 Final Exam W. Adams December 20,...

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MATH 405 Final Exam W. Adams December 20, 2002 Show all work. Justify all answers. This is an open book exam. 1. (20 points each) Let A = " 0 i - i 0 # . (a) Show that A is unitary. (b) Find a diagonal matrix D and a unitary matrix U such that A = UDU * . 2. (15 points each) Assume that the minimal polynomial, m A , and characteristic polyno- mial, P A of a matrix A are as given in the three cases below. Describe or write down all possible Jordan canonical (normal) forms for A . (a) P A ( t ) = ( t - 3) 4 ( t - 2) 3 and m A ( t ) = ( t - 3) 3 ( t - 2). (b) P A ( t ) = ( t - 2) 5 and m A ( t ) = ( t - 2) 3 . (c) P A ( t ) = ( t - λ 1 ) e 1 ( t - λ 2 ) e 2 ··· ( t - λ m ) e m and m A ( t ) = ( t - λ 1 ) e 1 - 1 ( t - λ 2 ) e 2 - 1 ··· ( t - λ m ) e m - 1 for integers e 1 2 ,...,e m 2. 3. (25 points) Find the Jordan canonical form of the matrix A = - 4 0 9 0 1 0 - 4 0 8 . 4. (25 points) Let V be a vector space of dimension 3 and let T be a linear operator on V . Assume that we have a basis v 1 ,v 2 ,v 3 of V such that the matrix of T with respect to this basis is the matrix A in the last problem. Show that 3 v 1 + 2 v 3 is an eigenvector
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This note was uploaded on 09/21/2009 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.

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