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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:
15A24Matrix equations and identities
15A06Linear equations (linear algebra)
15B33Matrices over special rings (quaternions, finite fields, etc.)
16E50Von Neumann regular rings and generalizations
15A09Matrix inversion, generalized inverses
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References:
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