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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 tex2html_wrap_inline5904 for all nonzero column vectors x. It is negative definite if tex2html_wrap_inline5908 for all nonzero x. It is positive semidefinite if tex2html_wrap_inline5912 and negative semidefinite if tex2html_wrap_inline5914 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 tex2html_wrap_inline5924 formed by the first i rows and columns of A. So, the first principal minor of A is the matrix tex2html_wrap_inline5932 , the second principal minor is the matrix tex2html_wrap_inline5934 , and so on.

To fix ideas, consider a tex2html_wrap_inline5948 symmetic matrix tex2html_wrap_inline5950 .

It is positive definite if:

det tex2html_wrap_inline5952
det tex2html_wrap_inline5954
and negative definite if:
det tex2html_wrap_inline5956
det tex2html_wrap_inline5954 .

It is positive semidefinite if:
det tex2html_wrap_inline5960
det tex2html_wrap_inline5962
and negative semidefinite if:
det tex2html_wrap_inline5964
det tex2html_wrap_inline5966 .


Michael A. Trick
Mon Aug 24 16:30:59 EDT 1998