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Ranks of the common solution to some quaternion matrix equations with applications. (English) Zbl 1337.15014
P. Bhimasankaram [Sankhyā, Ser. A 38, 404–409 (1976; Zbl 0411.15008)] has investigated the system of matrix equations \(AX = B\), \(XC = D\), and \(EXF = G\) over the complex number field \(\mathbb{C}\) and has given necessary and sufficient conditions for the existence of solutions.
The authors study the solutions of the system above over the quaternion algebra \(\mathbb{H}\). They first obtain explicit formulas on maximal and minimal ranks of four real matrices \(X_{1}\), \(X_{2}\), \(X_{3}\), and \(X_{4}\) in the quaternion solution \(X =X_{1}+X_{2}i+X_{3}j+X_{4}k\) of the system above over \(\mathbb{H}\). Furthermore, using these results, they give necessary and sufficient conditions for the matrix system over \(\mathbb{H}\) to have real and complex solutions.

MSC:
15A24 Matrix equations and identities
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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