A10 - AB is Hermitian if and only if AB = BA . 9. Unitarily...

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Math 235 Assignment 10 Not To Be Submitted 1. Consider C 3 with its standard inner product. Let ~ z = 1 + i 2 - i - 1 + i , ~w = 1 - i - 2 - 3 i - 1 . a) Evaluate h ~ z, ~w i and h ~w, 2 i~ z i . b) Find a vector in span { ~ z, ~w } that is orthogonal to ~ z . c) Write the formula for the projection of ~u onto S = span { ~ z, ~w } . 2. Let V be an inner product space, with complex inner product h , i . Prove that if < ~u,~v > = 0, then k ~u + ~v k 2 = k ~u k 2 + k ~v k 2 . Is the converse true? 3. Prove that for any n × n matrix A , we have det A = det A . 4. Let h , i denote the standard inner product on C n and let U be an n × n unitary matrix. a) Show that h U~ z,U ~w i = h ~ z, ~w i for any ~ z, ~w C n . b) Suppose λ is an eigenvalue of U . Use part a) to show that | λ | = 1. c) Find a unitary matrix U which has eigenvalue λ where λ 6 = ± 1. 5. Let A * = ( A ) T be the conjugate transpose of A . Prove that ( A * ) * = A , ( αA ) * = αA * and ( AB ) * = B * A * . 6. Let A be skew-Hermitian. Prove that all the eigenvalues of A are purely imaginary. 7. Let A be normal and invertible. Prove that B = A * A - 1 is unitary. 8. Let A and B be Hermitian matrices. Prove that
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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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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A10 - AB is Hermitian if and only if AB = BA . 9. Unitarily...

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