Constructive characterization of Lipschitzian Q₀-matrices

Abstract

A matrix M∈R^n×n has property (^∗∗) if M and all its principal pivotal transforms (PPTs) satisfy the property that the rows corresponding to the nonpositive diagonal entries are nonpositive. It has been shown that every Lipschitzian Q₀-matrix satisfies property (^∗∗). In this paper, it is shown that property (^∗∗) is also sufficient for a Lipschitzian matrix to be in Q₀. Property (^∗∗) has several consequences. If A has this property, then A and all its PPTs must be completely Q₀; further, for any q, the linear complementarity problem (q, A) can be processed by a simple principal pivoting method. It is shown that a negative matrix is an N-matrix if, and only if, it has property (^∗∗); a matrix is a P-matrix if, and only if, it has property (^∗∗) and its value is positive. Property (^∗∗) also yields a nice decomposition structure of Lipschitzian matrices. This paper also studies properties of Lipschitzian matrices in general; for example, we show that the Lipschitzian property is inherited by all the principal submatrices.