
Quadratic Forms
4
4.1 INTRODUCTION
Second-degree homogeneous expressions are
called quadratic forms. They occur in physics and
in geometry. In analytical geometry, for instance,
a quadratic form has to be transformed into its
principal-axes-form so as to determine the nature
of the conic section such as parabola, ellipse or
hyperbola, etc., if it involves two variables, and of
the quadratic surface such as paraboloid, ellipsoid
or hyperboloid, etc., if it involves three variables.
A quadratic form can be represented by
,1
n
ij i j
ij
Xaxx
=
=
∑
T
A
(4.1)
where X is a column n-vector and
A is a symmetric
matrix of the coeffi cients. We study here the method ...