Since det b 1 and det b is the product of the

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Unformatted text preview: √ 1/√2 −1/ 2 √ b1 = , b2 = . 1/ 2 1/ 2 The proof of the theorem of Section 2.1 shows that these vectors are orthogonal to each other, and hence the matrix √ √ 1/√2 −1/ 2 √ B= 1/ 2 1/ 2 is an orthogonal matrix such that 2 0 B T AB = 0 8 . Note that B represents a counterclockwise rotation through 45 degrees. If we define new coordinates (y1 , y2 ) by x1 x2 =B y1 y2 , equation (2.6) will simplify to y1 y2 or 2 2 2y1 + 8y2 = 2 0 0 8 y1 y2 = 1, 2 2 y1 y2 √ √ + = 1. 2 (1/ 2) (1/ 8)2 We recognize that this is the equation of √ ellipse. The lengths of the semimajor an √ and semiminor axes are 1/ 2 and 1/(2 2). 38 0.4 0.2 -0.4 -0.2 0.2 0.4 -0.2 -0.4 Figure 2.1: Sketch of the conic section 5x2 − 6x1 x2 + 5x2 − 1 = 0. 1 2 The same techniques can be used to sketch quadric surfaces in R 3 , surfaces defined by an equation of the form a11 a12 a13 x1 x1 x2 x3 a21 a22 a23 x2 = 1, a31 a32 a33 x3 where the aij ’s are constants. a11 A = a21 a31 If we let a12 a22 a32 a13 a23 , a33 x1 x = x2 , x3 we can write this in matrix form xT Ax = 1. (2.7) According to the Spect...
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This document was uploaded on 01/12/2014.

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