Page 340 - A Course in Linear Algebra with Applications

324 Chapter Nine: Advanced Topics

If X ^ 0, then BX ^ 0 since B is invertible. Hence \\BX\\ is
positive if X ^ 0. It follows that q, and hence A, is positive
definite.
Conversely, suppose that A is positive definite, so that
all its eigenvalues are positive. Now there is a real orthogonal
T
matrix S such that S AS = D is diagonal, with diagonal
entries d 1 ( ..., d n say. Here the di are the eigenvalues of A, so
all of them are positive. Define \[D to be the real diagonal
matrix with diagonal entries y/d~[,..., y/d^,. Then we have
1
l
T
A = {S )~ DS T = SDS T since S T = S" , and hence
T T
T
A = S(VDVD)S T = {y/DS ) {VDS ).
Finally, put B = \/DS T and observe that B is invertible since
both S and y/~D are.
Application to local maxima and minima
A well-known use of quadratic forms is to determine if
a critical point of a function of several variables is a local
maximum or a local minimum. We recall briefly the nature of
the problem; for a detailed account the reader is referred to a
textbook on calculus such as [18].
Let / be a function of independent real variables xi,...,
whose first order partial derivatives exist in some region
x n
R. A point P(ai, ...,a n) of R is called a local maximum (min-
imum) of / if within some neighborhood of P the function /
assumes its largest (smallest) value at P. A basic result states
that if P is a local maximum or minimum of f lying inside
R, then all the first order partial derivatives of f vanish at P:

f Xi(ai,...,a n) =0 for i = l,...,n.
A point at which all these partial derivatives are zero is called
a critical point of . Thus every local maximum or minimum
/
is a critical point of . However there may be critical points
/
which are not local maxima or minima, but are saddle points
of/.