8.5. Quadratic Forms

So far we have discussed only linear systems of equations and how to write these systems in matrix form. However, you should also be aware that quadratic expressions of the form

can be expressed as a matrix:

A quick check of the multiplication will verify that the matrix is equivalent to the given quadratic expression. The eigenvalues of this matrix define several important properties about the quadratic form including whether it is positive or negative definite or semidefinite or even hyperbolic.

Property	Zero	Min/Max	Eigenvalues
Positive Definite		min
Negative Definite		max
Positive Semidefinite		min	and
Negative Semidefinite		max	and
Hyperbolic	and	n/a	and

The column labeled "Zero" specifies where the function has a zero.

means a zero at the origin while

means that a zero exists somewhere other than the origin. For the hyperbolic quadratics,

and

means that there are values of

and

for which the function is positive some of the time and negative at others. The "Min/Max" column specifies whether the zero is a minimum or a maximum of the function. As for the eigenvalues,

as in the case of positive definite means that all the eigenvalues are positive. For the semidefinites, one eigenvalue is

and the other is either positive or negative. For hyperbolic quadratics, one eigenvalue is positive and the other is negative.

For example,

can be expressed as the following:

The eigenvalues of this matrix are

and

so this quadratic is positive semidefinite.

For quadratic forms of three variables, the equivalent matrix is given by the following:

Again, the same eigenvalue rules apply.