A note on E′-matrices

AbstractLet E∗ denote the class of square matrices M such that the linear complementarity problem Mz + q ⩾ 0, z ⩾ 0, (Mz + q)Tz = 0, has a unique solution for every q such that 0 ≠ q ⩾ 0. We show that E′ ≜ E∗ E, where E is the strictly semimonotone matrices, consists of completely Q0 matrices whose proper principal submatrices are completely Q matrices. We also show that (1) singular P1-matrices are in E∗ and those that are in E′ are U-matrices and (2) in the classes of adequate matrices and Z-matrices, the E′-matrices are precisely the singular P1-matrices that are not Q-matrices