A tensor is a multidimensional analog of a matrix. Q-matrices have been extensively studied in the context of the linear complementarity problem due to their solvability for any given vector q. In this article, we extend certain results of Q-matrices to Q-tensors. Characterizing a tensor as a Q-tensor remains a challenging problem in the literature. In this article, we establish sufficient conditions under which a principal subtensor of a Q-tensor is also a Q-tensor. Furthermore, we extend a result due to Huang, Suo and Wang. It is well-known that R-tensors are Q-tensors, although the converse does not always hold. We also provide conditions under which a Q-tensor can be classified as an R-tensor. Additionally, we prove several results pertaining to positive (nonnegative) tensors.

A matrix game is considered completely mixed if all the optimal pairs of strategies in the game are completely mixed. In this paper, we establish that a matrix game A, with a value of zero, is completely mixed if and only if the value of the game associated with A+Di is positive for all i, where Di represents a diagonal matrix where ith diagonal entry is 1 and else 0. Additionally, we address Kaplansky’s question from 1945 regarding whether an odd-ordered symmetric game can be completely mixed, and provide characterizations for odd-ordered skew-symmetric matrices to be completely mixed. Moreover, we demonstrate that if A is an almost skew-symmetric matrix and the game associated with A has value positive, then A+Di∈Q for all i, where Di is a diagonal matrix whose ith diagonal entry is 1 and else 0. Skew-symmetric matrices and almost skew-symmetric matrices with value positive fall under the class of P0 and Q0, making them amenable to processing through Lemke’s algorithm.

In 1945, Kaplansky [4] introduced the concept of the games being completely mixed and presented a necessary and sufficient condition for a game associated with a skew-symmetric matrix to be completely mixed. Recently, we have provided an additional condition for such games. It is known that skew symmetric matrices are Q0 and P0. In 1997, Murthy and Parthasarathy proved that if a matrix B belongs to fully copositive (C0f) and Q0, then B also belongs to P0. Building upon these results, our main result states that if the game associated with a fully copositive Q0-matrix B is completely mixed, then B+Dj∈Q for all j from 1 to n, where Dj is a diagonal matrix whose jth diagonal entry is 1 and else 0. Additionally, we prove that if B∈C0f∩Q0 but not a Q-matrix, then GB is completely mixed game if and only if B+Dj∈Q for all j from 1 to n.

In 1981, Stone conjectured that a fully semimonotone matrix is contained in In 1995, Murthy proved that for ifHere, we show that for matrices with some specific sign patterns this conjecture is true. Murthy showed that fully semimonotone Z-matrices are that is Here, we show that semimonotone Z-matrices are contained in that is, we exempt the condition of fully semimonotone with semimonotone. Further, we show the equivalency of -matrices and matrices for Z-matrices. Precisely, we are characterizing the matrices in These classes have been found to be interesting in view of the fact that these are processable by Lemke’s algorithm.

In 1979, Pang proved that within the class of semimonotone matrices, R-matrices are Q-matrices and conjectured that the converse is also true. Jeter and Pye gave a counterexample when n= 5 for the converse; namely, they gave a semimonotone matrix that is in Q but not in R. In this paper, we prove this conjecture for semimonotone matrices of order n≤ 3 and provide a counterexample when n> 3 , showing the sharpness of the result. We also provide an application of this result.