Page 343 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 327

matrix
••• f XlXn\
/f XlXl f XlXa
TT JX2X1 JX2X2 ' ' ' JX2X n

J Xf\X\ J XJIX2 J XfiXji
/
which is called the hessian of the function . Notice that the
f
hessian matrix is symmetric since f XiXj = Xj Xi, provided that
/ and all its derivatives of order < 3 are continuous.
The fundamental theorem may now be stated.
Theorem 9.2.6
Let f be a function of independent variables xi,... ,x n. As-
sume that f and its partial derivatives of order < 3 are contin-
uous in a region containing a critical point P(ai,a,2, • • • ,a n).
Let H be the the hessian of f.
/
(a) / H(ai,..., a n) is positive definite, then P is a local
minimum of f;
(b) if H(a±,..., o n ) is negative definite, then P is a local
maximum of f;
(c) if H(ai,..., a n) is indefinite, then P is a saddle point
off-
Example 9.2.5
2
Consider the function f(x, y) = (x — 2x) cos y. It has a
single critical point (1, n) since this is the only point where
both first derivatives vanish. To decide the nature of this point
we compute the hessian of / as

2 cos y — (2x — 2) sin y
H = 2
-{fix — 2) sin y —(x — 2x) cos y

Hence H(1,TT) — I J, which is clearly negative defi-

/
nite. Thus the point in question is a local maximum of .
Notice that the test given in 9.2.6 will fail to decide the
nature of the critical point P if at P the matrix H is not