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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.
15A24Matrix equations and identities
15A06Linear equations (linear algebra)
15A33Matrices over special rings
16E50Von Neumann regular rings and generalizations
15A09Matrix inversion, generalized inverses