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where P orthogonally diagonalizes A and .
EXAMPLE 6 Eliminating the Cross-Product Term
Describe the conic C whose equation is .
Solution
The matrix form of this equation is
(7)
where
The characteristic equation of A is
so the eigenvalues of A are and . We leave it for the reader to show that orthonormal bases for the eigenspaces are
Thus
orthogonally diagonalizes A. Moreover, , and thus the orthogonal coordinate transformation
is a rotation. Substituting 8 into 7 yields
Since (8)
this equation can be written as
or
which is the equation of the ellipse sketched in Figure 9.6.4. In that figure, the vectors and are the column vectors of
P—that is, the eigenvectors of A.
