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Characterization for the general solution to a system of matrix equations with quadruple variables. (English) Zbl 1354.15011

Summary: In this paper, we give some necessary and sufficient conditions for the solvability to the system of matrix equations \[ \begin{aligned} & A_1X_1=C_1, \quad X_2B_1=D_1\\ &A_2X_3=C_2,\quad X_3B_2=D_2,\\ &A_3X_4=C_3,\quad X_4B_3=D_3,\\ &A_4X_1+X_2B_4+C_4X_3D_4+C_5X_4D_5=E_1 \end{aligned} \eqno{(0.1)} \] and provide an expression of the general solution to (0.1). Furthermore, we obtain the maximal and minimal ranks of \(X_3\) and \(X_4\) in (0.1). The findings of this paper extend the known results in the literatures.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
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