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Unformatted text preview: ACS I Linear Algebra Homework # 3 1. Short proofs. a. A symmetric matrix has real eigenvalues. Use this fact to show that a symmetric positive definite matrix has positive eigenvalues, i.e., λ i > 0 for all i . b. Show how the SVD of an m × n matrix A with n linearly independent columns can be used to calculate its pseudo inverse A † = ( A T A ) 1 A T 2. Use Matlab (or some other library routine) to compute the SVD of the matrix A = 2 1 4 5 6 3 4 5 1 2 2 2 1 5 7 11 7 8 Use the SVD to answer the following questions. a. Give the rank of A and explain why it has this rank. Give the two condition number of A . b. Give vectors which form an orthonormal basis for the null space of A ; give vectors which form an orthonormal basis for the range of A . c. Determine a rank 1 approximation to A ; show how you got this. d. Determine a rank 2 approximation to A ; show how you got this....
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 Spring '11
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 Linear Algebra, linearly independent columns, Tridiagonal matrix, positive definite matrix, deﬁnite tridiagonal matrix, pseudo inverse A

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