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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 b-b a b) B = 4 i 1 + 3 i-1 + 3 i i c) C = 1 0 1 + i 2 1-i . 1 2 10. By direct computation, show that A = a b c d is a root of its characteristic polynomial. 11. Prove that if a matrix is unitary, Hermitian, or skew-Hermitian, then it is normal. Find a normal matrix that is not unitary, Hermitian or skew-Hermitian. 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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- Fall '08