solns8 - MATH 235/W08: Orthogonal Diagonalization,...

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Unformatted text preview: 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 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 parenrightbiggparenleftbigg ¯ a − 1 1 ¯ d parenrightbigg = parenleftbigg | ¯ a | 2 + 1 − a + ¯ d − ¯ a + d | ¯ d | + 1 parenrightbigg and A ∗ A = parenleftbigg ¯ a − 1 1 ¯ d parenrightbiggparenleftbigg 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 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 ....
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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solns8 - MATH 235/W08: Orthogonal Diagonalization,...

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