Ranks of the common solution to some quaternion matrix equations with applications (Q1954011)

\textit{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.