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9.6 In this section we shall show how to remove the cross-product terms from a
quadratic form by changing variables, and we shall use our results to study
DIAGONALIZING the graphs of the conic sections.
QUADRATIC FORMS;
CONIC SECTIONS

Diagonalization of Quadratic Forms

Let

(1)

be a quadratic form, where A is a symmetric matrix. We know from Theorem 7.3.1 that there is an orthogonal matrix P that
diagonalizes A; that is,

where , , …, are the eigenvalues of A. If we let

and make the substitution in 1, then we obtain
But

which is a quadratic form with no cross-product terms.
In summary, we have the following result.

THEOREM 9.6.1

Let be a quadratic form in the variables , , …, , where A is symmetric. If P orthogonally diagonalizes A, and if

the new variables , , … are defined by the equation , then substituting this equation in yields