Page 725 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

that orthogonally diagonalizes the matrix A. . This ensures that the orthogonal coordinate

Step 2. Interchange the columns of P, if necessary, to make
transformation

(5)

is a rotation. -system, substitute 5 into 4. This yields
Step 3. To obtain the equation for C in the

or (6)
Since P orthogonally diagonalizes A,

where and are eigenvalues of A. Thus 6 can be rewritten as

or

(where and ). This equation has no cross-product term.

The following theorem summarizes this discussion.

THEOREM 9.6.2

Principal Axes Theorem for
Let

be the equation of a conic C, and let

be the associated quadratic form. Then the coordinate axes can be rotated so that the equation
for C in the new -coordinate system has the form

where and are the eigenvalues of A. The rotation can be accomplished by the substitution