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A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

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Abstract

We consider the system of four linear matrix equations A 1 X = C 1, XB 2 = C 2, A 3 XB 3 = C 3 and A 4 XB 4 = C 4 over ℛ, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A 1 X = C 1 and A 3 X = C 3 to have a bisymmetric solution, the system of matrix equations A 1 X = C 1 and A 3 XB 3 = C 3 to have a perselfconjugate solution over ℛ with an involution and char ℛ ≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over ℛ are obtained. The previous known results on some systems of matrix equations are special cases of the new results.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

    Qing Wen Wang

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  1. Qing Wen Wang
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Correspondence to Qing Wen Wang.

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This research is supported by the Natural Science Foundation of China (No. 0471085), the Natural Science Foundation of Shanghai, the Development Foundation of Shanghai Educational Committee, and the Special Funds for Major Specialities of Shanghai Education Committee

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Wang, Q.W. A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications. Acta Math Sinica 21, 323–334 (2005). https://doi.org/10.1007/s10114-004-0493-1

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  • DOI: https://doi.org/10.1007/s10114-004-0493-1

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