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 \({\mathbf{{u}}} \ge \textbf{0},{\textbf{w}} = {\mathcal {A}}{\mathbf{{u}}}^{m-1}+\mathbf{{q}} \ge \textbf{0},~\text {and}~ {\mathbf{{u}}}^{T}{\textbf{w}} = 0\). 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 \({\mathcal {A}}\) of order m and dimension 2 with \({ rank({\mathcal {A}})=2}\), we show that \({\mathcal {A}}\) is an \(\textrm{R}_{0}\)-tensor. The idea is inspired by the recent work of Parthasarathy et al. (J Optim Theory Appl 195:131–147, 2022) and Sivakumar et al. (Linear Multilinear Algebra 70:6947–6964, 2021) on \({\mathrm Q}\)-matrices.

如果张量互补问题对所有\ ( \ mathbf { {q }} \in {{\mathbb {R}}^n}\)。这意味着对于每个向量\(\mathbf{{q}}\)，都存在一个向量\({\mathbf{{u}}}\)使得\({\mathbf{{u}}} \ge \textbf{0},{\textbf{w}} = {\mathcal {A}}{\mathbf{{u}}}^{m-1}+\mathbf{{q}} \ge \textbf{0 },~\text {和}~ {\mathbf{{u}}}^{T}{\textbf{w}} = 0\)。在本文中，我们证明在一阶对称张量类中，\({\mathrm Q}\) -张量正是正张量。此外，对于m阶、维度为 2的对称\({\mathrm Q}\) -张量\({\mathcal {A}}\)，其中 \({rank({\mathcal {A}})=2} \)，我们证明\({\mathcal {A}}\)是一个\(\textrm{R}_{0}\) -张量。这个想法的灵感来自于 Parthasarathy 等人最近的工作。(J Optim Theory Appl 195:131–147, 2022) 和 Sivakumar 等人。（线性多重线性代数 70:6947–6964, 2021）关于\({\mathrm Q}\)矩阵。