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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 1 XB 1 =C 1 ,A 2 XB 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)
15A33Matrices over special rings
16E50Von Neumann regular rings and generalizations
15A09Matrix inversion, generalized inverses