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Zbl 1176.15020
Wang, Qingwen; Song, Guangjing; Liu, Xin
Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications.
(English)
[J] Algebra Colloq. 16, No. 2, 293-308 (2009). ISSN 1005-3867

The first author studied the system of linear matrix equations $A_1X = C_1, XB_2 = C_2, A_3XB_3 = C_3$ and $A_4XB_4 = C_4$ over a von Neumann regular ring $R$ with unity [{\it Q. Wang}, Acta Math. Sin., Engl. Ser. 21, No. 2, 323--334 (2005; Zbl 1083.15021)]. \par In this paper, the authors establish the formulas of the maximal and minimal ranks of the common solution of the linear matrix equations $A_1X = C_1, XB_2 = C_2, A_3XB_3 = C_3$ and $A_4XB_4 = C_4$ 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.
[Sheng Chen (Harbin)]
MSC 2000:
*15A24 Matrix equations
15A03 Vector spaces
15A09 Matrix inversion
15A33 Matrices over special rings

Keywords: system of matrix equations; division ring; maximal rank; minimal rank; generalized inverses

Citations: Zbl 1083.15021

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