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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. Unitarily diagonalize the following matrices. a) A = a b- b a Solution: We have C ( λ ) = λ 2- 2 aλ + a 2 + b 2 so by the quadratic formula we get eigenvalues λ = a ± bi . For λ = a + bi we get A- λI =- bi b- b- bi ⇒ ~ z 1 = 1 i For λ = a- bi we get A- λI = 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 # . b) B = 4 i 1 + 3 i- 1 + 3 i i Solution: We have C ( λ ) = λ 2- 5 iλ + 6 so by the quadratic formula we get eigenvalues λ = 6 i and λ =- i . For λ = 6 i we get B- λI...
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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.
- Fall '08