A Criterion for $\mathrm{Q}$-tensors

Abstract

A tensor $\mathcal{A}$ of order $m$ and dimension $n$ is called a $\mathrm{Q}$-tensor if the tensor complementarity problem has a solution for all $\mathbf{q}\in {\mathbb{R}}^n$. This means that for every vector $\mathbf{q}$, there exists a vector $\mathbf{u}$ such that $\mathbf{u}\ge 0,\mathbf{w}=\mathcal{A}{\mathbf{u}}^{m-1}+\mathbf{q}\ge 0,\text{and}{\mathbf{u}}^T\mathbf{w}=0$. In this paper, we prove that within the class of rank one symmetric tensors, the $\mathrm{Q}$-tensors are precisely the positive tensors. Additionally, for a symmetric $\mathrm{Q}$-tensor $\mathcal{A}$ with $rank(\mathcal{A})=2$, we show that $\mathcal{A}$ is an ${\mathrm{R}}_0$-tensor. The idea is inspired by the recent work of Parthasarathy et al. \cite{Parthasarathy} and Sivakumar et al. \cite{Sivakumar} on $\mathrm{Q}$-matrices.