Positive Definite Matrices

When we study functions of several variables (see Chapter 3!), we will need the following matrix notions.

A square matrix A is positive definite if for all nonzero column vectors x. It is negative definite if for all nonzero x. It is positive semidefinite if and negative semidefinite if for all x. These definitions are hard to check directly and you might as well forget them for all practical purposes.

More useful in practice are the following properties, which hold when the matrix A is symmetric (that will be the case of interest to us), and which are easier to check.

The ith principal minor of A is the matrix formed by the first i rows and columns of A. So, the first principal minor of A is the matrix , the second principal minor is the matrix , and so on.

• The matrix A is positive definite if all its principal minors , have strictly positive determinants.
• If these determinants are nonzero and alternate in signs, starting with det( , then the matrix A is negative definite.
• If the determinants are all nonnegative, then the matrix is positive semidefinite,
• If the determinant alternate in signs, starting with det( , then the matrix is negative semidefinite.

To fix ideas, consider a symmetic matrix .

It is positive definite if:

(i)

det

(ii)

det

and negative definite if:

(i)

det

(ii)

det .

It is positive semidefinite if:

(i)

det

(ii)

det

and negative semidefinite if:

(i)

det

(ii)

det .