Q<inf>#</inf>-matrices and Q<inf>†</inf>-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 (Formula presented.). This means that for every vector q, there exists a vector x ≥ 0 such that y = Ax + q ≥ 0 and x T y = 0. Two new classes of matrices are studied, namely the Q #-matrices and (Formula presented.) -matrices. If for every vector q ∈ R(A), there exists a vector x ∈ R(A) satisfying x ≥ 0, y = Ax + q ≥ 0 and x T y = 0, then A is a Q #-matrix. If the vector x satisfying the above properties is instead required to be in R(A T), then A is a (Formula presented.) -matrix. Properties of these classes of matrices and their relationship with the class of Q-matrices are studied.