M235A8 - MATH 235 Assignment 8 Orthogonal Diagonalization...

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Unformatted text preview: MATH 235 Assignment 8 Orthogonal Diagonalization, Symmetric & Complex Matrices Hand in questions 1,3,5,7,9,11 by 9:30 am on Wednesday November 22, 2006. 1. (a) Let A,B be n × n matrices and suppose B = A- 1 A T and B is symmetric. Prove that A 2 is symmetric. (b) Suppose C is a real n × n matrix such that C is symmetric and C 2 = C and let D = I n- 2 C with I n denoting the n × n identity matrix. Prove that D is symmetric and orthogonal. 2. (a) Find all complex 2 × 2 matrices A = [ a ij ] which are both unitary and Hermitian, and have a 11 = 1 / 2. (b) Show that a real 2 × 2 normal matrix is either symmetric or has the form " a b- b a # ( a,b ∈ R ). 3. Let Z be an m × n complex matrix such that Z * Z = I n where I n denotes the n × n identity matrix. (a) Show that H = ZZ * is Hermitian and satisfies H 2 = H . (b) Show that U = I n- 2 ZZ * is both unitary and Hermitian. 4. Let A be a complex n × n matrix. (a) Show that A = H + K for some Hermitian matrix H and some skew-Hermitian matrix K ....
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M235A8 - MATH 235 Assignment 8 Orthogonal Diagonalization...

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