Page 337 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 321

2 2
—2x — 3y is clearly negative definite. On the other hand,
the form 2x 2 — 3y 2 can take both positive and negative values,
so it is indefinite.
In these examples it was easy to decide the nature of the
quadratic form since it contained only squared terms. How-
ever, in the case of a general quadratic form, it is not possible
to decide the nature of the form by simple inspection. The
diagonalization process for symmetric matrices allows us to
reduce the problem to a quadratic form whose matrix is diag-
onal, and which therefore involves only squared terms. From
this it is apparent that it is the signs of the eigenvalues of the
matrix A that are important. The definitive result is

Theorem 9.2.2
T
Let A be a real symmetric matrix and let q = X AX: then
(a) q is positive definite if and only if all the eigenvalues
of A are positive;
(b) q is negative definite if and only if all the eigenvalues
of A are negative;
(c) q is indefinite if and only if A has both positive and
negative eigenvalues.

Proof
T
There is a real orthogonal matrix S such that S AS = D
1
is diagonal, with diagonal entries c\,..., c n, say. Put X =
T
S X; then X = SX' and
T
T
T
q = X AX = (X') (S AS)X' = (x'fDX,
so that q takes the form

q = c\x' 2 + c 2x' 2 2 H h c nx' n 2

where the entries of X . Thus q, considered as
a quadratic form in x'-y,...,x' n, involves only squares. Now
observe that as X varies over the set of all non-zero vectors