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Reducible solution to a quaternion tensor equation

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Abstract

We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation \(\mathscr{A}{ * _N}\mathscr{X}{ * _N}\mathscr{B} = \mathscr{C}\) 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 \({\mathscr{A}_1}{ * _N}{\mathscr{X}_1}{ * _M}{\mathscr{B}_1} + {\mathscr{A}_1}{ * _N}{\mathscr{X}_2}{ * _M}{\mathscr{B}_2} + {\mathscr{A}_2}{ * _N}{\mathscr{X}_3}{ * _M}{\mathscr{B}_2} = \mathscr{C}\), which plays a key role in investigating the reducible solution to \(\mathscr{A}{ * _N}\mathscr{X}{ * _N}\mathscr{B} = \mathscr{C}\). 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.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11971294).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Mengyan Xie & Qing-Wen Wang

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  1. Mengyan Xie
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  2. Qing-Wen Wang
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Correspondence to Qing-Wen Wang.

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Xie, M., Wang, QW. Reducible solution to a quaternion tensor equation. Front. Math. China 15, 1047–1070 (2020). https://doi.org/10.1007/s11464-020-0865-6

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