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M235A4b - MATH 235 Diagonalization 2 Assignment 4b Hand in...

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MATH 235 Diagonalization - 2 Assignment 4b Hand in questions 2,3,4,5,6,7,9 by 9:30 am on Wednesday October 25, 2006. 1. Find the eigenvalues and eigenvectors of A = 1 1 0 0 4 1 0 0 a 0 2 0 c 0 b 2 . Is A diagonalizable? 2. Diagonalize, if possible, the following matrices: A = 1 0 1 0 1 0 1 0 1 , B = - 2 4 i 4 i - 4 i 2 0 4 i 0 2 . 3. A 3 × 3 real matrix A has eigenvalues λ 1 = λ 2 = - 2 , λ 3 = 3 and associated eigenvectors v 1 = 1 0 1 , v 2 = 0 1 1 , v 3 = 1 1 1 , respectively. Find A . 4. Consider the matrix: A = 0 2 0 0 k 0 2 0 0 k 0 2 0 0 k 0 where k is a constant. (a) Find a value of k such that A is diagonalizable. (b) Find a value of k such that A is not diagonalizable. 5. For A = 4 - 6 3 - 5 , evaluate A n for arbitrary positive integer n . 6. Let A, B M n × n ( R ), we say that A and B are simultaneously diagonalizable (S.D.) if there exists an invertible matrix Q in M n × n ( R ) such that Q - 1 AQ = D 1 and Q - 1 BQ = D 2 where D
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