Page 709 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Thus, subject to the constraint , the maximum value of the quadratic form is , which occurs if ,

; and the minimum value is , which occurs if , . Moreover, alternative bases for

the eigenspaces can be obtained by multiplying the basis vectors above by −1. Thus the maximum value, , also occurs if
.
, ; similarly, the minimum value, , also occurs if ,

DEFINITION

A quadratic form is called positive definite if for all , and a symmetric matrix A is called a positive
definite matrix if
is a positive definite quadratic form.

The following theorem is an important result about positive definite matrices.

THEOREM 9.5.2

A symmetric matrix A is positive definite if and only if all the eigenvalues of A are positive.

Proof Assume that A is positive definite, and let be any eigenvalue of A. If x is an eigenvector of A corresponding to ,

then and , so

(7)

where is the Euclidean norm of x. Since it follows that , which is what we wanted to

show. for all . But if , we can
Conversely, assume that all eigenvalues of A are positive. We must show that

normalize x to obtain the vector with the property . It now follows from Theorem 9.5.1 that

where is the smallest eigenvalue of A. Thus,

Multiplying through by yields
which is what we wanted to show.

EXAMPLE 3 Showing That a Matrix Is Positive Definite