Page 338 - A Course in Linear Algebra with Applications

322 Chapter Nine: Advanced Topics

n T T 1
in R , so does X' = S X. This is because S = S' is
invertible. Therefore q > 0 for all non-zero X if and only if
q > 0 for all non-zero X . In this way we see that it is sufficient
to discuss the behavior of q as a quadratic form in x[,..., x' n.
Clearly q will be positive definite as such a form precisely when
c\,..., c n are all positive, with a corresponding statement for
negative definite: but q is indefinite if there are positive and
negative Q'S. Finally ci,..., c n are just the eigenvalues of A,
so the assertion of the theorem is proved.

Let us consider in greater detail the important case of a
quadratic form q in two variables x and y, say
2
q = ax 2 + 2bxy + cy ; the associated symmetric matrix is

Let the eigenvalues of A be d± and d 2. Then by 8.1.3 we have
the relations det(A) = d\d 2 and tr(A) = d\ + d 2; hence

2
d\d 2 = ac — b and d\ + d 2 = a + c.
Now according to 9.2.2 the form q is positive definite if and
only if d\ and d 2 are both positive. This happens precisely
when ac > b 2 and a > 0. For these conditions are certainly
necessary if d\ and d 2 are to be positive, while if the conditions
hold, a and c must both be positive since the inequality ac > b 2
shows that a and c have the same sign.
In a similar way we argue that the conditions for A to be
2
negative definite are ac > b and a < 0. Finally, q is indefinite
2
if and only if ac < b : for by 9.2.2 the condition for q to be
indefinite is that d\ and d 2 have opposite signs, and this is
equivalent to the inequality d\d 2 < 0. Therefore we have the
following result.