# 2 1 2 1 82 we recognize that this is the equation of

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Unformatted text preview: ctral Theorem from Section 2.1, there exists a 2 × 2 orthogonal matrix B of determinant one such that B −1 AB = B T AB = If we make a linear change of variables, x = B y, 36 λ1 0 0 λ2 . then since xT = yT B T , equation (2.3) is transformed into yT (B T AB )y = 1, y1 0 λ2 λ1 0 y2 y1 y2 = 1, or equivalently, 2 2 λ1 y1 + λ2 y2 = 1. (2.5) In the new coordinate system (y1 , y2 ), it is easy to recognize the conic section: • If λ1 and λ2 are both positive, the conic is an ellipse . • If λ1 and λ2 have opposite signs, the conic is an hyperbola . • If λ1 and λ2 are both negative, the conic degenerates to the the empty set , because the equation has no real solutions. In the case where λ1 and λ2 are both positive, we can rewrite (2.5) as 2 y1 ( 1/λ1 )2 + 2 y2 = 1, ( 1/λ2 )2 from which we recognize that the semi-major and semi-minor axes of the ellipse are 1/λ1 and 1/λ2 . The matrix B which relates x and y represents a rotation of the plane. To see this, note that the ﬁrst column b1 of B is a...
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