Q#-matrices and Q†-matrices: two extensions of the Q-matrix concept

A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a solution for all q∈Rn. This means that for every vector q, there exists a vector x≥0 such that y=Ax+q≥0 and xTy=0. Two new classes of matrices are studied, namely the Q#-matrices and Q†-matrices. If for every vector q∈R(A), there exists a vector x∈R(A) satisfying x≥0, y=Ax+q≥0 and xTy=0, then A is a Q#-matrix. If the vector x satisfying the above properties is instead required to be in R(AT), then A is a Q†-matrix. Properties of these classes of matrices and their relationship with the class of Q-matrices are studied.