Unformatted text preview: ral Theorem, there is a 3 × 3 orthogonal matrix B of
determinant one such that B T AB is diagonal. We introduce new coordinates y1
x = B y,
y = y2 y3
and equation (2.7) becomes
yT (B T AB )y = 1.
Thus after a suitable linear change of coordinates, the equation (2.7) can be put
in the form λ1 0
y1 y2 y3 0 λ2 0 y2 = 1,
2 -4 Figure 2.2: An ellipsoid.
λ1 y1 + λ2 y2 + λ3 y3 = 1, where λ1 , λ2 , and λ3 are the eigenvalues of A. It is relatively easy to sketch the
quadric surface in the coordinates (y1 , y2 , y3 ).
If the eigenvalues are all nonzero, we ﬁnd that there are four cases:
• If λ1 , λ2 , and λ3 are all positive, the quadric surface is an ellipsoid .
• If two of the λ’s are positive and one is negative, the quadric surface is an
hyperboloid of one sheet .
• If two of the λ’s are negative and one is positive, the quadric surface is an
hyperboloid of two sheets .
• If λ1 , λ2 , and λ3 are all negative, the equation represents the empty set .
Just as in the cas...
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- Winter '14