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

Keywords:

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