Reducible solution to a quaternion tensor equation

Abstract

We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation \mathcal{A}*\mathcal{N}\mathcal{X}*\mathcal{N}\mathcal{B}=\mathcal{C}via Einstein product using Moore-Penrose inverse, and present an expression of the reducible solution to the equation when it is solvable. Moreover, to have a general solution, we give the solvability conditions for the quaternion tensor equation {\mathcal{A}}_{1*\mathcal{N}}{\mathcal{X}}_{1*\mathcal{M}}{\mathcal{B}}_1+{\mathcal{A}}_{1*\mathcal{N}}{\mathcal{X}}_{2*\mathcal{M}}{\mathcal{B}}_2+{\mathcal{A}}_{2*\mathcal{N}}{\mathcal{X}}_{3*\mathcal{M}}{\mathcal{B}}_2=\mathcal{C}, which plays a key role in investigating the reducible solution to \mathcal{A}*\mathcal{N}\mathcal{X}*\mathcal{N}\mathcal{B}=\mathcal{C}. The expression of such a solution is also presented when the consistency conditions are met. In addition, we show a numerical example to illustrate this result.

Key words： Quaternion tensor; quaternion tensor equation; Einstein product; Moore-Penrose inverse; general solution; reducible solution

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