Page 344 - A Course in Linear Algebra with Applications

328 Chapter Nine: Advanced Topics

positive definite, negative definite or indefinite: for example,
H might equal 0 at P.

Extremal values of a quadratic form
Consider a quadratic form in variables x±,..., x n

T
q = X AX,

where as usual A is a real symmetric nxn matrix and X is the
column consisting of x±,..., x n. Suppose that we want to find
the maximum and minimum values of q when X is subject to
a restriction. One possible restriction is that

||X|| = a

2
for some a > 0, that is, x\ + • • • + x\ — a . Thus we are
looking for the maximum and minimum values of q on the n-
n
sphere with radius a and center the origin in R . One could
use calculus to attack this problem, but it is simpler to employ
diagonalization.
There is a real orthogonal nxn matrix S such that
T
S AS = D, where D is the diagonal matrix with the eigen-
1
values of A, say di,..., d n, on its diagonal. Put Y = S~ X:
thus we have X = SY and

T
T T
2
T
q = X AX = Y S ASY = Y DY = d lV\ + ••• + d ny n,
where yi,..., y n are the entries of Y.
In addition we find that

T
T
T
T
X X = Y (S S)Y = Y Y,
T _ 1
since S = 5 . Therefore our problem may be reformulated
as follows: find the maximum and minimum values of the ex-
2
pression diyf-\ \-dnVn subject to y 2 + - • -+yn = a . But this