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Determinantal expression of the general solution to a restricted system of quaternion matrix equations with applications

Guang-Jing Song
대한수학회 (The Korean Mathematical Society)|2018.07|대한수학회보 (Bulletin of the KMS)|제55권 제4호|pp. 1285-1301

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키워드

determinantquaternion matrixCramer's rule
주제분류수학

초록

In this paper, we mainly consider the determinantal representations of the unique solution and the general solution to the restricted system of quaternion matrix equations \[ \left\{ \begin{array} [c]{c} A_{1}X=C_{1}\\ XB_{2}=C_{2}, \end{array} \right. \mathcal{R}_{r}\left( X\right) \subseteq T_{1},\text{ } \mathcal{N}_{r}\left( X\right) \supseteq S_{1}, \] respectively. As an application, we show the determinantal representations of the general solution to the restricted quaternion matrix equation \[ AX+YB=E,\text{ }\mathcal{R}_{r}\left( X\right) \subseteq T_{1},\text{ }\mathcal{N}_{r}\left( X\right) \supseteq S_{1},\text{ }\mathcal{R} _{l}\left( Y\right) \subseteq T_{2},\text{ }\mathcal{N}_{l}\left( Y\right) \supseteq S_{2}. \] The findings of this paper extend some known results in the literature.

Abstract

In this paper, we mainly consider the determinantal representations of the unique solution and the general solution to the restricted system of quaternion matrix equations \[ \left\{ \begin{array} [c]{c} A_{1}X=C_{1}\\ XB_{2}=C_{2}, \end{array} \right. \mathcal{R}_{r}\left( X\right) \subseteq T_{1},\text{ } \mathcal{N}_{r}\left( X\right) \supseteq S_{1}, \] respectively. As an application, we show the determinantal representations of the general solution to the restricted quaternion matrix equation \[ AX+YB=E,\text{ }\mathcal{R}_{r}\left( X\right) \subseteq T_{1},\text{ }\mathcal{N}_{r}\left( X\right) \supseteq S_{1},\text{ }\mathcal{R} _{l}\left( Y\right) \subseteq T_{2},\text{ }\mathcal{N}_{l}\left( Y\right) \supseteq S_{2}. \] The findings of this paper extend some known results in the literature.

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