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Unformatted text preview: AB is Hermitian if and only if AB = BA . 9. Unitarily diagonalize the following matrices. a) A = ± a bb a ² b) B = ± 4 i 1 + 3 i1 + 3 i i ² c) C = 1 0 1 + i 2 1i . 1 2 10. By direct computation, show that A = ± a b c d ² is a root of its characteristic polynomial. 11. Prove that if a matrix is unitary, Hermitian, or skewHermitian, then it is normal. Find a normal matrix that is not unitary, Hermitian or skewHermitian. 12. Let A be an n × n matrix. If A 2 = A , show that rank A = tr A . 13. Let A and B be n × n matrices such that AB = BA . a) Prove that A and B have a common eigenvector. b) Prove that there exists a unitary matrix U such that U * AU = T 1 and U * BU = T 2 where T 1 and T 2 are upper triangular....
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 Spring '10
 WILKIE
 Math, Matrices, CN, Orthogonal matrix, Normal matrix, Hermitian matrix

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