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On the existence of a common solution \(X\) to the matrix equations \(A_{i} XB_{j}\)=\(C_{ij}\), \((i,j){\in}\Gamma\). (English) Zbl 1037.15014

The problem of a common solution to a set of matrix equations \[ A_iXB_j=C_{ij}, \quad (i,j) \in \Gamma , \tag \(*\) \] where \(A_i, B_j\), \(C_{ij}\) are given matrices, \(X\) is an unknown matrix and \(\Gamma\) denotes a set of index pairs \((i,j)\); \(i=1,2, \dots, k\); \(j=1,2, \dots, k\); is investigated. (The matrices \(A_i, B_j\), \(C_{ij}\) and \(X\) have suitable dimensions). Necessary and sufficient conditions (in terms of the matrices and without use of Kronecker products) are established for the existence of a common solution to a set of matrix equations \((*)\).

MSC:

15A24 Matrix equations and identities
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