A10_soln - Math 235 Assignment 10 Solutions 1. Let A be...

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Unformatted text preview: Math 235 Assignment 10 Solutions 1. Let A be skew-Hermitian. Prove that all the eigenvalues of A are purely imaginary. Solution: If A is skew-Hermitian, then A * =- A . Hence ( iA ) * =- iA * = iA so iA is Hermitian. Then U * ( iA ) U = D and so U * AU = iD . 2. Let A be normal and invertible. Prove that B = A * A- 1 is unitary. Solution: We have A * A = AA * . So B * B = ( A * A- 1 ) * ( A * A- 1 ) = ( A- 1 ) * AA * A- 1 = ( A- 1 ) * A * AA- 1 = I. 3. Let A and B be Hermitian matrices. Prove that AB is Hermitian if and only if AB = BA . Solution: If AB is Hermitian, then ( AB ) = ( AB ) * = B * A * = BA, since A and B are Hermitian. If AB = BA , then ( AB ) * = B * A * = BA = AB. 4. Unitarily diagonalize the following matrices. a) A = a b- b a Solution: We have C ( ) = 2- 2 a + a 2 + b 2 so by the quadratic formula we get eigenvalues = a bi . For = a + bi we get A- I =- bi b- b- bi ~ z 1 = 1 i For = a- bi we get A- I = bi b- b bi ~ z 2 =- 1 i Hence we have D = a + bi a- bi and U = " 1 2- 1 2 i 2 i 2 # . 1 2 b) B = 4 i 1 + 3 i- 1 + 3 i i Solution: We have C ( ) = 2- 5 i + 6 so by the quadratic formula we get eigenvalues = 6 i and =- i . For = 6 i we get B- I =- 2 i 1 + 3 i- 1 + 3...
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This note was uploaded on 04/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

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A10_soln - Math 235 Assignment 10 Solutions 1. Let A be...

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