homework_6 - A-1 . 5. In each of the following, if...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Homework 6 Name: Student No.: Class: Final Result: I acknowledge that I am aware of University policy and regulations on hon- esty in academic work, and of the disciplinary guidelines and procedures appli- cable to breaches of such policy and regulations, as contained in the website http://www.cuhk.edu.hk/policy/academichonesty/ Signature Date Answer all the questions. 1. Compute the QR factorization of A (a) A = 1 2 - 1 - 2 1 1 (b) A = 1 0 2 - 1 2 0 - 1 - 2 2 2. Find the least squares line and a quadratic least squares polynomial for given data (3 , 2) , (4 , 3) , (5 , 2) , (6 , 4) , (7 , 3) 3. Find the eigenvalues and the corresponding eigenspaces for each of the following matrices. (a) A = 2 3 0 0 1 0 0 0 2 (b) A = 2 2 2 4 0 2 3 2 0 0 1 1 0 0 0 1 4. Let A be a nonsingular matrix and let λ be an eigenvalue of A . Show that 1 is an eigenvalue of
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Unformatted text preview: A-1 . 5. In each of the following, if possible, factor the matrix A into a product XDX-1 , where D is diagonal (a) A = • 4 2 3 3 ‚ (b) A = 1 0-1 3 3 2-2 (c) A = 2-2 3 3-2-1 2 2 6. For each of the matrices in Question 5, use the XDX-1 factorization to compute A 6 . 7. For each of the nonsingular matrices in Question 5, use use the XDX-1 factorization to compute A-1 8. In each of the following, if possible, orthogonally diagonalize each given matrix A , giving the diagonal matrix D and the diagonalizing orthogonal matrix P . (a) A = • 2 2 2 2 ‚ (b) A = -1-1-1-1-1-1 9. Show that if A is an n × n orthgonal matrix and x and y are vectors in R n , then ( A x ) · ( A y ) = x · y . 10. Show that if A is an orthogonal matrix, then A-1 is also orthogonal....
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homework_6 - A-1 . 5. In each of the following, if...

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