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Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications. (English) Zbl 1176.15020

The first author studied the system of linear matrix equations ${A}_{1}X={C}_{1},X{B}_{2}={C}_{2},{A}_{3}X{B}_{3}={C}_{3}$ and ${A}_{4}X{B}_{4}={C}_{4}$ over a von Neumann regular ring $R$ with unity [Q. Wang, Acta Math. Sin., Engl. Ser. 21, No. 2, 323–334 (2005; Zbl 1083.15021)].

In this paper, the authors establish the formulas of the maximal and minimal ranks of the common solution of the linear matrix equations ${A}_{1}X={C}_{1},X{B}_{2}={C}_{2},{A}_{3}X{B}_{3}={C}_{3}$ and ${A}_{4}X{B}_{4}={C}_{4}$ over an arbitrary division ring. Corresponding results in some special cases are also given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of the results in this paper.

##### MSC:
 15A24 Matrix equations and identities 15A03 Vector spaces, linear dependence, rank 15A09 Matrix inversion, generalized inverses 15A33 Matrices over special rings