| A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications |
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| 作者姓名： | QingWenWANG |
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| 作者单位： | DepartmentofMathematics,ShanghaiUniversity,Shanghai200444,P.R.China |
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| 摘 要： | We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, 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 A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results.
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| 关 键 词： | 冯诺伊曼正则环 矩阵方程 中心对称矩阵 共轭矩阵 |
| 收稿时间： | 26 August 2003 |
A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications |
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| QingWenWANG.A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications[J].Acta Mathematica Sinica,2005,21(2):323-334. |
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| Authors: | Qing Wen Wang |
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| Institution: | (1) Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China |
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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.
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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| Keywords: | von Neumann regular ring System of matrix equations Perselfconjugate matrix Centrosymmetric matrix Bisymmetric matrix |
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