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THEOREM 9.7.1
Principal Axes Theorem for
Let
be the equation of a quadric Q,and let
be the associated quadratic form. The coordinate axes can be rotated so that the equation of Q
in the -coordinate system has the form
where , , and are the eigenvalues of A. The rotation can be accomplished by the
substitution
where P orthogonally diagonalizes A and det .
EXAMPLE 3 Eliminating Cross-Product Terms
Describe the quadric surface whose equation is
Solution (4)
The matrix form of the above quadratic equation is
where
As shown in Example 1 of Section 7.3, the eigenvalues of A are and , and A is orthogonally diagonalized by the
matrix
where the first two column vectors in P are eigenvectors corresponding to , and the third column vector is an
eigenvector corresponding to .
Since (verify), the orthogonal coordinate transformation is a rotation. Substituting this expression in 4
yields
