Karamardian Matrices: A Generalization of $Q$-Matrices

A real square matrix $A$ is called a $Q$-matrix if the linear complementarity problem $LCP(A,q)$ has a solution for all $q \in \mathbb{R}^n$. This means that for every vector $q$ there exists a vector $x$ such that $x \geq 0, y=Ax+q\geq 0$ and $x^Ty=0$. A well known result of Karamardian states that if the problems $LCP(A,0)$ and $LCP(A,d)$ for some $d\in \mathbb{R}^n, d >0$ have only the zero solution, then $A$ is a $Q$-matrix. By relaxing the condition on $d$ and imposing a condition on the solution vector $x$ in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, requiring that these two modified problems have only zero as a solution. In this article, a systematic treatment of Karamardian matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of $Q$-matrices. A subclass of a recently introduced notion of $P_{\#}$-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of $P_{\#}$-matrices and make some fundamental contributions.