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Determinantal Representations of the Solutions to Systems of Generalized Sylvester Equations

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Abstract

In this paper, we consider three systems of coupled generalized Sylvester quaternion equations

$$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1} \\ A_{2}X_{2}-Y_{1}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}, \right. \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2}\\ A_{3}Y_{2}-Y_{3}B_{3}=C_{3} \end{array}, \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}. \right. \end{aligned}$$

We present new necessary and sufficient conditions for the solvability of each system, and derive the determinantal representations of the general solutions to the above systems by the row and column determinants of quaternion matrices.

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Acknowledgements

This research is supported by The Science and Technology Development Fund, Macau SAR (File no. 185/2017/A3), the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN 312386-2015) and UM URGP, National Natural Science Foundation of China (11571220) and (11801354).

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Authors and Affiliations

  1. Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, 999078, Macau, People’s Republic of China

    Xin Liu

  2. School of Mathematics and Information Sciences, Weifang University, Weifang, 261061, People’s Republic of China

    Guang-Jing Song

  3. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Yang Zhang

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  1. Xin Liu
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  2. Guang-Jing Song
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  3. Yang Zhang
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Correspondence to Guang-Jing Song.

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Communicated by G. Stacey Staples

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Liu, X., Song, GJ. & Zhang, Y. Determinantal Representations of the Solutions to Systems of Generalized Sylvester Equations. Adv. Appl. Clifford Algebras 30, 12 (2020). https://doi.org/10.1007/s00006-019-1038-0

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