Constructive characterization of Lipschitzian Q0-matrices Original Research Article

A matrix M set membership, variant 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.