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Common solutions of a pair of matrix equations. (English) Zbl 1020.15013

The author studies the simultaneous solvability of the matrix equations \(AX+XB=M\) (the Sylvester equation) and \((*)\) \(AXB=C\). He gives necessary and sufficient conditions for the solvability of the couple and when these conditions are met he gives a formula for the general solution. He shows that the common solution \(X_0\) of the couple is unique if and only if \(A\) and \(B\) are nonsingular and \(AC+CB=AMB\); in this case one has \(X_0=A^{-1}CB^{-1}\). He also gives necessary and sufficient conditions for the solvability of the couple of equations \(A_1X+XB_1=M\) and \((*)\). Applications to generalized inverses \(A^-\) (i.e. solutions to \(AXA=A\)) are given.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
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