WEAK COLUMN SUFFICIENT TENSORS AND THEIR APPLICATIONS

Abstract

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