Weak column sufficient tensors and their applications

Column sufficient matrices have wide applications in mathematical analysis, linear complementarity problems, etc., since they contain many important special matrices, such as Hilbert matrices, B-matrices, double B-matrices, H-matrices with nonnegative diagonal entries, P-matrices, M-matrices, and positive (semi-)definite matrices, etc. Column sufficient tensors have recently arisen in connection with the tensor complementarity problem (TCP). By modifying the existing definitions of column sufficient tensors which have some defects for odd order tensors, we propose a formula for the definition of (weak) column sufficient tensor. The proposed column sufficient tensor classes coincide with the existing ones of even orders. We show that all Z-eigenvalues of an even-order symmetric weak column sufficient tensor are nonnegative, and all its H-eigenvalues are nonnegative, regardless of whether the order is even or odd. In contrast to the LCP theory, when the involving tensor in the tensor complementarity problem is a column sufficient tensor, by giving a counterexample, we show that the solution set of the tensor complementarity problem is not necessarily convex. After that several results on tensor complementarity problems are established.