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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.

##### MSC:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.) 93B25 Algebraic methods 93C05 Linear systems in control theory
##### Keywords:
matrix equation; principal ideal domain
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