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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 semimajor and semiminor 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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 Winter '14
 Equations

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