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Math_235-Assig_7-F09

Math_235-Assig_7-F09 - Math 235 Assignment 7 Due Wednesday...

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Unformatted text preview: Math 235 Assignment 7 Due: Wednesday, Nov 11th 1. For each quadratic form Q(:E’), determine the corresponding symmetric matrix A. By diagonalizing A, express QC?) in diagonal form and give an orthogonal matrix that diago- nalizes A. Classify each quadratic form. ._MW-.,[email protected]_= 7:52 + 124,9 M- 7. mm- a W a m , WWW b) 62(93, 3/) = $2 + 6561/ - 7'112 c) Q(ac, y, z) 2 23:2 + 2y2 — 322 - 4mg + 63w + 63/2. 2. Sketch the graph of each of the following equations showing both the original and new axes. For any hyperbola, find the equation of the asymptotes. a) 7:52 + 12mg + 12y2 = 48. b) $2 + 6mg — 7y2 x 32. b c a) Prove that Q is positive definite if det A > 0 and a > 0. b) Prove that Q is negative definite if det A > O and a < 0. c) Prove that Q is indefinite if det A < 0. 3. Let Q($,y) :2 :‘ETAi‘ with A = [a b] and detA 7é O. 4. Let < , > be an inner product on R” and let S x {51, . . . ,é’n} be the standard basis. 71 TL a) Verify that < 53,175: E Z $341k < 535:5; > for any £5,176 R”. ‘=1k::1 b) Let G be the n x 71 matrix defined by gjk =< é}, é’k > and verify that < 53,37 5: CE'TGQ’. c) Show that there is a basis B =2 {171, . . . ,Un} such that in B—coordinates, < 13,17 >= Alflgl + Azfgfiz + ‘ ‘ ' + Anfngn, where A1, . . . ,/\n are the eigenvalues of G. d) Introduce a new basis 0 = {1171,. . . ,u‘in} by defining 111} = ”D’j/fl //\j. Use an asterisk to denote Clcoordinates7 so that :E’ z 2?wa + - - - + £71117”. 1 if j =2 k, 0 if j % k ’ Thus, with respect to the inner produce < ,5, C is an orthonormal basis, and in C‘— coordinates, the inner product of two vectors looks just like the standard dot product. Verify that < raj, wk >= { and that < 63327 >= 9211/1“ + - - - + fly; 5. Let A be an n X n symmetric matrix and let :5, 176 R". Define < 11?, 17 >: fTAg]. Prove " " that ' Z ; 5 "is afi'rfih’e’f' product onR’f’iT’andonlyif Aispositive"definite: " ' if? Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems, and submit written answers to the questions along with the rest of your assignment. Quadratic Forms For each quadratic form Q(x): 0 Determine the corresponding symmetric matrix A. o By diagonalizing A, express Q(x) in diagonal form and give an orthogonal matrix that diagonalizes A. o Classify Q(x) as positive definite, negative definite, or neither. (a) £200 (b) Q(x) = 085(2012 +$22 + $32 +3342) — 0.1x15v2 + 0.6:v1m3 +0.2m15v4 + 0.2m2x3 +0.6:r2m4 — 0. 1.713.“ —0.1:r12 + 2.1.T22 +1.33%2 +1.12%2 — 0.8:r1m2 +1.2m1x4 + 1.61‘2333 + 4.2333334 ...
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