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A system of real quaternion matrix equations with applications. (English) Zbl 1180.15019
A system of general linear matrix equations over the real quaternions is studied and solvability conditions with explicit expressions of the solutions, when the matrix system is solvable, are given. Let $${\mathbb H}^{n \times m}$$ stands for the set of all $$n \times m$$ matrices over the real quaternion algebra. A matrix $$A \in {\mathbb H}^{n \times m}$$ is called $$(P,Q)$$-symmetric or $$(P,Q)$$-skew-symmetric if $$A=PAQ$$ or $$A= -PAQ$$, respectively, where $$P \in {\mathbb H}^{n \times n}$$, $$Q \in {\mathbb H}^{m \times m}$$ are involutory matrices, i. e., $$P^2 = I$$, $$Q^2 = I$$. Firstly, some new necessary and sufficient conditions for the existence of a solution to the system
\begin{aligned} A_1 X_1 &= C_1, \qquad A_2 X_2 = C_3,\\ X_1 B_1 &= C_2, \qquad X_2 B_2 = C_4, \qquad A_3 X_1 B_3 + A_4 X_2 B_4 = C_5 \end{aligned}
over $${\mathbb H}$$ are established and an explicit expression of the general solution is derived. Some auxiliary results on certain related systems over $${\mathbb H}$$ are also given. Then, as application, necessary and sufficient conditions for the existence of the $$(P, Q)$$-symmetric solution to the system
$A_a X = C_a, \qquad X B_b = C_b, \qquad A_c X B_c = C_c$
over $${\mathbb H}$$, which was first investigated by P. Bhimasankaram [Sankhyā Ser. A 38, 404–409 (1976; Zbl 0411.15008)], are given. Finally, an algorithm and a numerical example to illustrate the presented results are shown.

##### MSC:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15B57 Hermitian, skew-Hermitian, and related matrices 15A09 Theory of matrix inversion and generalized inverses
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