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Math 235 Assignment 10 Not To Be Submitted 1. Let A be skew-Hermitian. Prove that all the eigenvalues of A are purely imaginary. 2. Let A be normal and invertible. Prove that B = A * A - 1 is unitary. 3. Unitarily diagonalize the following matrices. a) A = ± a b - b a ² b) B = ± 4 i 1 + 3 i - 1 + 3 i i ² c) C = 1 0 1 + i 0 2 0 1 - i 0 0 . 4. By direct computation, show that A =
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Unformatted text preview: a b c d is a root of its characteristic polynomial. 5. 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. 6. If A 2 = A , show that rank A = tr A . 1...
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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