Linear Algebra Solutions 79

Linear Algebra Solutions 79 - xy + 8 y 2 = X t AX , where A...

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x 2 y 2 8 16 -8 -16 8 16 -8 -16 x y x 2 y 2 0.95 1.9 2.85 -0.95 -1.9 -2.85 0.95 1.9 2.85 -0.95 -1.9 -2.85 x y Figure 2: (a): 4 xy - 3 y 2 = 8; (b): 8 x 2 - 4 xy + 5 y 2 = 36 Then X t AX = 4 x 2 1 + 9 y 2 1 and the original equation 8 x 2 - 4 xy + 5 y 2 = 36 becomes 4 x 2 1 + 9 y 2 1 = 36, or the standard form x 2 1 9 + y 2 1 4 = 1 , which represents an ellipse as in Figure 2(b). The axes of symmetry turn out to be y = 2 x and x = - 2 y . 4. We give the sketch only for parts (i), (iii) and (iv). We give the working for (ii) only. See Figures 3(a) and 4(a) and 4(b), respectively. (ii) We have to investigate the equation 5 x 2 - 4 xy + 8 y 2 + 4 5 x - 16 5 y + 4 = 0 . (3) Here 5 x 2 - 4
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Unformatted text preview: xy + 8 y 2 = X t AX , where A = 5-2-2 8 and X = x y . The eigenvalues of A are the roots of 2-13 +36 = 0, namely 1 = 9 and 2 = 4. Corresponding unit eigenvectors turn out to be [1 / 5 ,-2 / 5] t and [2 / 5 , 1 / 5] t . Hence if P = " 1 5 2 5-2 5 1 5 # , then P is an orthogonal matrix. Also as det P = 1, P is a proper orthogonal matrix and the equation x y = P x 1 y 1 82...
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