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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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