Page 720 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

where , , …, are the eigenvalues of A and

The matrix P in this theorem is said to orthogonally diagonalize the quadratic form or reduce the quadratic form to a sum of
squares.

EXAMPLE 1 Reducing a Quadratic Form to a Sum of Squares to a sum of squares, and express the
Find a change of variables that will reduce the quadratic form
quadratic form in terms of the new variables.

Solution

The quadratic form can be written as

The characteristic equation of the matrix is

so the eigenvalues are , , . We leave it for the reader to show that orthonormal bases for the three
eigenspaces are

Thus, a substitution that eliminates cross-product terms is

or, equivalently,
The new quadratic form is