P-matrices and N-matrices

Abstract

In this chapter we will give a geometric characterization of P-matrices. We will give some properties of N-matrices. These facts we need later to prove global univalence results due to Gale, Nikaido and Inada. We will also see the interrelation between P-matrices and positive quasi-definite matrices. Finally we examine the question whether P-property holds good under multiplication (sum) of two P-matrices — this kind of result is useful in determining when the composition F o G (sum, F+G) of two univalent functions is univalent.

Let A be an n×n matrix with entries real numbers. We will not consider matrices with complex entries. If A is a symmetric matrix then A is positive definite if the associated quadratic form x′Ax>0, for any x different from 0. Here prime denotes the transpose of the vector x. It is well known that a symmetric matrix A is positive definite if and only if every principal minor of A is positive. Suppose we drop the symmetric assumption from A. In such situations can we prove similar results? In other words, suppose A has the following property, namely x′Ax>0 for every x≠0. They can we assert that every principal minor of A is positive? Another interesting question is to characterize matrices whose principal minors are positive. Next we will answer these questions.