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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_inline1918 for all nonzero column vectors x. It is negative definite if tex2html_wrap_inline1922 for all nonzero x. It is positive semidefinite if tex2html_wrap_inline1926 and negative semidefinite if tex2html_wrap_inline1928 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_inline1938 formed by the first i rows and columns of A. So, the first principal minor of A is the matrix tex2html_wrap_inline1946 , the second principal minor is the matrix tex2html_wrap_inline1948 , and so on.

To fix ideas, consider a tex2html_wrap_inline1962 symmetic matrix tex2html_wrap_inline1964 .

It is positive definite if:

det tex2html_wrap_inline1966
det tex2html_wrap_inline1968
and negative definite if:
det tex2html_wrap_inline1970
det tex2html_wrap_inline1968 .

It is positive semidefinite if:
det tex2html_wrap_inline1974
det tex2html_wrap_inline1976
and negative semidefinite if:
det tex2html_wrap_inline1978
det tex2html_wrap_inline1980 .


Michael A. Trick
Mon Aug 24 13:24:14 EDT 1998