In this paper, we investigate the least-squares solution with the least norm to the following system of tensor equations over quaternions $\begin{array}{rl}& {\mathcal{A}}_1{\ast }_N\mathcal{X}={\mathcal{D}}_1,\mathcal{Y}{\ast }_N{\mathcal{B}}_2={\mathcal{D}}_2,{\mathcal{A}}_3{\ast }_N\mathcal{X}{\ast }_N{\mathcal{B}}_3={\mathcal{D}}_3,\\ & {\mathcal{A}}_4{\ast }_N\mathcal{Y}{\ast }_N{\mathcal{B}}_4={\mathcal{D}}_4,{\mathcal{A}}_5{\ast }_N\mathcal{X}+\mathcal{Y}{\ast }_N{\mathcal{B}}_5={\mathcal{D}}_5,\end{array}$ where $\mathcal{X},\mathcal{Y}$ are unknown quaternion tensors and the others are given quaternion tensors. Using the expressions of the Moore-Penrose inverses of partitioned tensors, we give a representation of the solution and an algorithm to compute this solution.

中文翻译：

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四元数代数上张量方程组的最小范数的最小二乘解

在本文中，我们研究了以下四元数张量方程组的最小范数的最小二乘解$\begin{array}{rl}& {\mathcal{一个}}_1{*}_{ñ}\mathcal{X}={\mathcal{D}}_1,\mathcal{是}{*}_{ñ}{\mathcal{乙}}_2={\mathcal{D}}_2,{\mathcal{一个}}_3{*}_{ñ}\mathcal{X}{*}_{ñ}{\mathcal{乙}}_3={\mathcal{D}}_3,\\ & {\mathcal{一个}}_4{*}_{ñ}\mathcal{是}{*}_{ñ}{\mathcal{乙}}_4={\mathcal{D}}_4,{\mathcal{一个}}_5{*}_{ñ}\mathcal{X}+\mathcal{是}{*}_{ñ}{\mathcal{乙}}_5={\mathcal{D}}_5,\end{array}$在哪里$\mathcal{X},\mathcal{是}$是未知的四元数张量，其他的是四元数张量。使用分区张量的 Moore-Penrose 逆表达式，我们给出了解的表示和计算该解的算法。