Constructive characterization of Lipschitzian Q0-matrices

AbstractA matrix M ∈ Rn × 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