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### hw7

Course: MATH 571, Fall 2008
School: Michigan
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Word Count: 322

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571 Math (Winter 2009): Homework 7 Due Date: See web page [i] In the class, we discussed how to nd the best rank-1 approximation to an m n matrix using the SVD. Describe how to nd the best rank-2 approximation. Explain your answer. [ii] Let A be a square matrix. Then its exponential is dened as exp(A) = I + A + A2 /2! + A3 /3! + If A = QT Q is the Schur factorization of A, show that exp(A) = Q exp(T )Q . If A...

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571 Math (Winter 2009): Homework 7 Due Date: See web page [i] In the class, we discussed how to nd the best rank-1 approximation to an m n matrix using the SVD. Describe how to nd the best rank-2 approximation. Explain your answer. [ii] Let A be a square matrix. Then its exponential is dened as exp(A) = I + A + A2 /2! + A3 /3! + If A = QT Q is the Schur factorization of A, show that exp(A) = Q exp(T )Q . If A = V DV 1 is the eigendecomposition of A (with D being diagonal), show that exp(A) = V exp(D)V 1 . [iii] Let x be an m-dimensional vector and let A be an m m matrix. In the class, we learnt how to solve the linear system dx/dt = Ax using the the eigenvalues and eigenvectors of A. Show how to solve the linear system dx = Ax + f (t), dt where f (t) is an m-dimensional vector that is a function of t. Express the solution to that linear system compactly using the matrix [iv] exponential. Consider the eigenvalue problem d2 u = u, dx2 with 0 x and u(0) = u() = 0. Solve the eigenvalue problem analytically to nd innitely many eigenvalues . If the interval 0 x is divided into a grid with spacing h = /m, then there are grid points at u(ih) for i = 0, 1, . . . , m. The second derivative at ih can be approximated as u((i + 1)h) 2 u(ih) + u((i 1)h) , h2 for i = 1, 2, . . . , m 1. In addition, we know that u(ih) = 0 for i = 0 and i = m. Use the grid approximation to turn the continuous eige...

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