Page 330 - A Course in Linear Algebra with Applications

314 Chapter Nine: Advanced Topics

where the a^- are real numbers. Setting A = [ar,] n)Tl and
writing X for the column vector with entries x\,..., x n, we see
from the definition of matrix products that q may be written
in the form
T
q = X AX.
Thus the quadratic form q is determined by the real matrix
A.
At this point we make the crucial observation that noth-
T
ing is lost if we assume that A is symmetric. For, since X AX
T
T T
is scalar, q may also be written as (X AX) T = X A X;
therefore

T
T
T
T T
q= \{X AX + X A X) = X { \{A + A ) )X.
ZJ Zi
It follows that A can be replaced by the symmetric matrix
T
^(A + A ). For this reason it will in future be tacitly assumed
that the matrix associated with a quadratic form is symmetric.
The observation of the previous paragraph allows us to
apply the Spectral Theorem to an arbitrary quadratic form.
The conclusion is that a quadratic form can be written in
terms of squares only.

Theorem 9.2.1
T
Let q = X AX be an arbitrary quadratic form. Then there is
a real orthogonal matrix S such that q = cix[ + • • • + c nx' n
T
where x[,..., x' n are the entries of X — S X and C\,..., c n
are the eigenvalues of the matrix A.

Proof
T
By 9.1.3 there is a real orthogonal matrix S such that S AS =
D is diagonal, with diagonal entries c±,..., c n say. Define X
T
to be S X; then X = SX . Substituting for X, we find that
T
T
T
T
q = X AX = (SX') A(SX') = {X') {S AS)X'
T
= (X') DX'.