Page 339 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 323

Corollary 9.2.3
2 2
Let q = ax + 2bxy + cy be a quadratic form in x and y.
Then:
(a) q is positive definite if and only if ac > b 2 and a > 0;
(b) q is negative definite if and only if ac > b 2 and a < 0;
2
(c) q is indefinite if and only if ac < b .
Example 9.2.3
2
Let q = -2x 2 + xy - 3y . Here we have a = - 2 , b = 1/2,
c = —3. Since ac — b 2 > 0 and a < 0, the quadratic form is
negative definite, by 9.2.3.
The status of a quadratic form in three or more variables
can be determined by using 9.2.2.

Example 9.2.4
Let q = —2x 2 — y 2 — 2z 2 + 6xz be a quadratic form in x, y, z.
The matrix of the form is

1-2 0 3
A= 0 - 1 0
\ 3 0 - 2

which has eigenvalues —5, —1,1. Hence q is indefinite.
Next we record a very different criterion for a matrix to
be positive definite. While it is not a practical test, it has a
very striking form.
Theorem 9.2.4
Let A be a real symmetric matrix. Then A is positive definite
T
if and only if A — B B for some invertible real matrix B.
Proof
T
Suppose first that A = B B with B an invertible matrix.
T
Then the quadratic form q = X AX can be rewritten as

2
T
T T
q = X B BX = (BX) BX = \\BX\\ .