## 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.)

Zbl 0411.15008