homework2 - Copy

homework2 - Copy - M is symmetric M = M • Prove that the...

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HOMEWORK #2 DUE WEDNESDAY, JULY 14 STATISTICS 132, SUMMER 2010 Question 1: HTF 3.3 (b). You may use without proof the result of part (a) of HTF 3.3. Question 2: HTF 3.6 Question 3: HTF 3.7 Question 4: HTF 3.9 Question 5: HTF 3.12 Question 6: HTF 3.17. Set aside 20% of your data as test data. You may omit the final row and the final column of Table 3.3. Include R code. Question 7: HTF 3.29 Question 8: If M is an n by n matrix, a vector x is said to be a (right) eigenvector of M if (1) M x = λ x for some constant λ . If (1) holds, λ is said to be an eigenvalue of M corresponding to the eigenvector x . Suppose that a n by n matrix M with real entries has n distinct eigenvalues λ 1 > λ 2 ... > λ n , and suppose that
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Unformatted text preview: M is symmetric ( M = M ). • Prove that the eigenvectors x 1 ,..., x n corresponding to λ 1 ,...,λ n are orthogonal. Hint: consider x i M x j-x j M x i . • Show that λ 1 = max z : || z || =1 z M z and λ 2 = max z : || z || =1 z x 1 =0 z M z . Hint: since there are n orthogonal eigenvectors, we can normalize these to find an orthonormal basis of eigenvectors. Start by writing z as a linear combination of these basis elements. • Comment on the relevance of these observations to equation (3.49) and the sur-rounding discussion in HTF. Extra Credit: HTF 3.30 1...
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This note was uploaded on 07/14/2010 for the course STAT 132 taught by Professor Haulk during the Spring '10 term at UBC.

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