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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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