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554-HW-7q - 4 7 Let T be the linear operator on R which is...

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7 Let T be the linear operator on R 4 which is represented in the standard ordered basis by the matrix M = 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c 0 Under what conditions on a, b, c is T diagonalizable? 7 Let T be a linear operator on the n -dimensional vector space V , and suppose that T has n distinct characteristic values. Prove that T is diagonalizable. 7 Let A and B F n × n . Prove that if ( I - AB ) is invertible then ( I - BA ) is invertible with inverse I + B ( I - AB ) - 1 A . 7 Prove for A, B F n × n that AB and BA have precisely the same characteristic values in F . 7 Suppose that A R 2 × 2 is symmetric. Prove that A is similar over R to a diagonal matrix. 7 Let N C 2 × 2 such that N 2 = 0. Prove that either N = 0 or N is similar over C to 0 0 1 0 . 7 If A C 2 × 2 show that A is similar over C to either a diagonal matrix or a matrix of the form s 0 1 s . 7 Let V = F n × n . Let A be a fixed element of V . Let T be the linear operator on V given by left multiplication by the matrix A . Is it true that A and T have the same characteristic values?
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