## A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity.(English)Zbl 1058.15015

Many problems in systems and control theory require the solution of Sylvester’s equation $$AX-YB=C$$ or of its generalization $$(*)$$ $$AXB+CYD=E$$. The author studies the couple of matrix equations $$(**)$$ $$A_1XB_1=C_1,A_2XB_2=C_2$$ over an arbitrary regular ring with identity. He obtains necessary and sufficient conditions for the consistency of the system $$(**)$$ and presents its general solution. The results are used to obtain necessary and sufficient conditions for the consistency of the equation $$(*)$$ and to derive the form of its general solution.

### MSC:

 15A24 Matrix equations and identities 15A06 Linear equations (linear algebraic aspects) 15B33 Matrices over special rings (quaternions, finite fields, etc.) 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 15A09 Theory of matrix inversion and generalized inverses
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