Associated with a given symmetric matrix , we can construct quadratic form where is an any non-zero vector. The matrix is said to be

*positive definite*, if*positive semi-definite*, if*negative definite*, if*negative semi-definite*, if

For example, consider the covariance matrix of a random vector

The corresponding quadratic form is

where is a scalar. Therefore is positive semi-define.

iff all of its eigenvalues are greater than zero:

As the eigenvalues of are , we have iff .

Ruye Wang 2014-08-15