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A system of four matrix equations over von Neumann regular rings and its applications. (English) Zbl 1083.15021
The author considers the following matrix systems over a von Neumann regular ring \(R\) with unity (i.e. for each \(a\in R\) there exists \(b\in R\) such that \(aba=a\)): (*) \(A_1X=C_1\), \(XB_2=C_2\), \(A_3XB_3=C_3\), \(A_4XB_4=C_4\); (**) \(A_1X=C_1\), \(A_3X=C_3\); (***) \(A_1X=C_1\), \(A_3XB_3=C_3\). He gives necessary and sufficient conditions for solvability, and he explicits the general solution to system (*), the bisymmetric one to system (**) and the perselfconjugate one to system \((***)\), the latter two in the case when there is an involution \(\sigma\) over \(R\) (char\(R\neq 2\)). An \(n\times n\)-matrix \(A=(a_{ij})\) is bisymmetric if \(a_{ij}=a_{n-i+1,n-j+1}=\sigma (a_{ji})\); it is perselfconjugate if \(A=A^{(*)}\) where \(A^{(*)}=VA^*V\), \(V\) being the permutation matrix with units along the antidiagonal and zeros elsewhere, and \(A^*=(\sigma (a_{ji}))\).

MSC:
15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices
15A09 Theory of matrix inversion and generalized inverses
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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