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solns8 - MATH 235/W08 Orthogonal Diagonalization Symmetric...

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MATH 235/W08: Orthogonal Diagonalization, Symmetric & Complex Matrices, Solutions 8 April 2, 2008. Contents 1 Properties of Symmetric/Hermitian/Normal Matrices*** 2 2 More on Hermitian/Unitary Matrices 2 3 Hermitian, Orthogonal Projections*** 3 4 Hermitian and Skew-Hermitian Parts 3 5 Quadratic Forms*** 3 6 Normal Matrices 4 7 Orthogonal Diagonalization*** 4 8 Eigenspaces 5 9 Unitary Diagonalization*** 5 10 Symmetric Square Root 6 11 Orthogonal Eigenvectors*** 6 12 Common Eigenpairs 6 13 MATLAB*** 7 13.1 Colliding Eigenvalues*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 13.2 Equation of an Orbit*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 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? BEGIN SOLUTION: If A = A , then AA = A ( A ) and also A A = ( A ) A = A 2 , the same value. END SOLUTION. 2. Why is every unitary matrix normal? BEGIN SOLUTION: If AA = I , then A = A 1 is the unique inverse of A , i.e. A A = I and AA = I , the same value. END SOLUTION. 3. For what values of a, d is the 2 × 2 matrix parenleftbigg a 1 1 d parenrightbigg normal? BEGIN SOLUTION: Let A = parenleftbigg a 1 1 d parenrightbigg . Then AA = parenleftbigg a 1 1 d parenrightbigg parenleftbigg ¯ a 1 1 ¯ d parenrightbigg = parenleftbigg | ¯ a | 2 + 1 a + ¯ d ¯ a + d | ¯ d | + 1 parenrightbigg and A A = parenleftbigg ¯ a 1 1 ¯ d parenrightbigg parenleftbigg a 1 1 d parenrightbigg = parenleftbigg | ¯ a | 2 + 1 d + ¯ a ¯ d + a | ¯ d | + 1 parenrightbigg . Therefore, for A to be normal we must have d + ¯ a = a + ¯ d, ¯ d + a = ¯ a + d. Equivalently, d + ¯ d = a + ¯ a. We conclude that a, d must have equal real parts. END SOLUTION. 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. 2
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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 . BEGIN SOLUTION: First, note that ZZ is Hermitian for any matrix Z since ( ZZ ) = ( Z ) Z = ZZ . Moreover, H 2 = ( ZZ )( ZZ ) = Z ( Z Z ) Z = ZZ = H . (Remark: This shows that H is a Hermitian idempotent . An idempotent ( H 2 = H ) means that H is a projection. And, a Hermitian idempotent means that this is an orthogonal projection.) END SOLUTION. 2. Show that U = I n 2 ZZ is both unitary and Hermitian.
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