HW13-solutions

# HW13-solutions - huang(dh34953 HW13 gilbert(54160 This...

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huang (dh34953) – HW13 – gilbert – (54160) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Every symmetric matrix is orthogonally di- agonalizable. True or ±alse? 1. ±ALSE 2. TRUE correct Explanation: An n × n matrix A is orthogonally diagonal- izable iF and only iF A is a symmetric matrix. Thus, any symmetric matrix must be diago- nalizable. Consequently, the statement is TRUE . 002 10.0 points A symmetric n × n A matrix always has n distinct real eigenvalues. True or ±alse? 1. TRUE 2. correct Explanation: A symmetric n × n matrix A always has n real eigenvalues, but they need not be distinct. ±or example, when A = 3 2 4 2 6 2 4 2 3 , then det[ A λI ] = λ 3 + 12 λ 2 21 λ 91 = ( λ 7) 2 ( λ + 2) . Thus this 3 × 3 matrix A has only 2 distinct eigenvalues, 7 and 2. Consequently, the statement is . 003 10.0 points When u 1 = b 2 1 B , u 2 = b 1 2 B , are eigenvectors oF a symmetric 2 × 2 matrix A corresponding to eigenvalues λ 1 = 2 , λ 2 = 12 , fnd matrices D and P in an orthogonal diag- onalization oF A . 1. D = b 2 0 0 12 B , P = b 1 2 2 1 B 2. D = b 2 0 0 12 B , P = b 2 1 1 2 B 3. D = b 2 0 0 12 B , P = 1 5 b 2 1 1 2 B correct 4. D = b 12 0 0 2 B , P = b 2 1 1 2 B 5. D = b 12 0 0 2 B , P = 1 5 b 2 1 1 2 B 6. D = b 2 0 0 12 B , P = 1 5 b 1 2 2 1 B Explanation: When D = b λ 1 0 0 λ 2 B , Q = [ u 1 u 2 ] , then Q has orthogonal columns and A = QDQ - 1 is a diagonalization oF A , but

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huang (dh34953) – HW13 – gilbert – (54160) 2 it is not an orthogonal diagonalization be- cause Q is not an orthogonal matrix. We have to normalize u 1 and u 2 : set v 1 = u 1 b u 1 b = 1 5 b 2 1 B , v 2 = u 2 b u 2 b = 1 5 b 1 2 B . Then P = [ v 1 v 2 ] is an orthogonal matrix and so A = PDP - 1 is an orthogonal diagonalization of A when D = b 2 0 0 12 B , P = 1 5 b 2 1 1 2 B . 004 10.0 points A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix. True or False? 1. FALSE 2. TRUE correct Explanation: If the matrix of a quadratic form is a di- agonal matrix, then the quadratic form has no cross-product terms. Conversely, if a quadratic form has no cross-product terms, then the matrix of the quadratic form is a diagonal matrix. Consequently, the statement is TRUE . 005 10.0 points The expression b x b 2 is a quadratic form. True or False? 1. TRUE correct 2. Explanation: The identity matrix I is symmetric because it’s diagonal. So Q ( x ) = x T I x = x T x = b x b 2 is a quadratic form. Consequently, the statement is TRUE . 006 10.0 points If A is symmetric, then the change of vari- able x = P y transforms Q ( x ) = x T A x into a quadratic form with no cross-product term for any orthogonal matrix P . True or False?
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