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The equation \(AXB + CYD = E\) over a principal ideal domain. (English) Zbl 0742.15006
The paper is concerned with the solvability of the matrix equation \(AXB+CYD=E\), where the entries of the matrices belong to a principal ideal domain \(R\). After a certain reduction, the statement is that the equation is solvable if and only if certain two block matrices constructed from \(A\), \(B\), \(C\), \(D\), \(E\), and zero matrices are equivalent over \(R\) with respect to elementary transformations of the rows and columns. The author also relates the set of solutions to the solution sets of other equations.

15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
93B25 Algebraic methods
93C05 Linear systems in control theory
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