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Unformatted text preview: Math 235 Assignment 8 Due: Wednesday, July 6th 1. For each quadratic form QUE), determine the corresponding symmetric matrix A. By
diagonalizing A, express Q65) in diagonal form and give an orthogonal matrix that diago—
nalizes A. Classify each quadratic form. a) Q(:23, y) 2 7:232 + 12am +12y2.
b) 62(56, 2/) = 3:2 + 6:239  7y2
c) Q(:c, y, z) 2 2372 + 2y2 —— 322 — 4333; + 63% + 6yz. 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) 7562 + 12333; +12y2 = 48.
b) 52:2 + Gary — 7y2 = 32. 3. Let Q(:23, y) r: ETAEE with A = [a b] and detA # 0. b c
a) Prove that Q is positive deﬁnite if det A > 0 and a > 0.
b) Prove that Q is negative deﬁnite if detA > 0 and a < 0.
c) Prove that Q is indeﬁnite if detA < 0. 4. Let A be an n X n symmetric matrix and let 53’, 3] E R”. Deﬁne < 55, 3] >2: ﬂAg’. Prove
that < , > is an inner product on R” if and only if A is positive deﬁnite. 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 To ﬁnd the eigenvalues of a matrix in MATLAB, use the eig command. 11 ~10 4
For example, the eigenvalues of the matrix A 2 12 ~15 8 can be found as follows:
12 —18 11 >> A = [11 1O 4; 12 15 8; 12 ~18 11]
>> eigCA) MATLAB returns that the eigenvalues of A are 5, 1, and 3. For the quadratic form Q(X) =3 33612 ~ 4.11332 ‘ 7.1342 + 511352 + 2271.932 ‘1' 4951324 ‘ 421311175 ~ 6.1132333 ‘l" 8332334 ‘l" 251335134 ~ 2$3$5, (a) What is the corresponding symmetric matrix A? Enter it into MATLAB. (b) Find the orthogonal matrix, R, that diagonalizes A and the corresponding
diagonal matrix, D. There are a couple of ways to ﬁnd a basis for each eigenspace; consider how you might use
the rref command (as you would on paper) or investigate the options of the eig command. (c) Express Q<CI§> in diagonal form. (d) Classify Q(:c) as positive deﬁnite, negative deﬁnite, or neither. ...
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 Spring '08
 CELMIN

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