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# assign8 - MATH 235/W08 Orthogonal Diagonalization Symmetric...

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MATH 235/W08: Orthogonal Diagonalization, Symmetric & Complex Matrices, Assignment 8 Hand in questions 1,3,5,7,9,11,13 by 9:30 am on Wednesday April 2, 2008. Contents 1 Properties of Symmetric/Hermitian/Normal Matrices*** 2 2 More on Hermitian/Unitary Matrices 2 3 Hermitian, Orthogonal Projections*** 2 4 Hermitian and Skew-Hermitian Parts 2 5 Quadratic Forms*** 2 6 Normal Matrices 3 7 Orthogonal Diagonalization*** 3 8 Eigenspaces 3 9 Unitary Diagonalization*** 3 10 Symmetric Square Root 3 11 Orthogonal Eigenvectors*** 4 12 Common Eigenpairs 4 13 MATLAB*** 5 13.1 Colliding Eigenvalues*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 13.2 Equation of an Orbit*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1

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1 Properties of Symmetric/Hermitian/Normal Matri- ces*** A (complex) normal matrix is defined by A * A = AA * ; it has orthogonal eigenvectors. A skew- Hermitian matrix is defined by A * = - A . 1. Why is every skew-Hermitian matrix normal? 2. Why is every unitary matrix normal? 3. For what values of a,d is the 2 × 2 matrix parenleftbigg a 1 - 1 d parenrightbigg normal? 2 More on Hermitian/Unitary Matrices 1. Let A,B be n × n matrices and suppose B = A - 1 A T and B is symmetric. Prove that A 2 is symmetric. 2. 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. 3. Find all complex 2 × 2 matrices A = [ a ij ] which are both unitary and Hermitian, and have a 11 = 1 / 2. 3 Hermitian, Orthogonal Projections*** Let Z be an m × n complex matrix such that Z * Z = I n where I n denotes the n × n identity matrix. 1. Show that H = ZZ * is Hermitian and satisfies H 2 = H . 2. Show that U = I n - 2 ZZ * is both unitary and Hermitian.
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