Geodesic
The matrix equation $AX-YB=C$ and related problems
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 15-23

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The main result of the paper is a theorem, using which a new proof of Roth's theorem is obtained, a new solvability criterion for the matrix equation $AX-YB=C$ is proved, a formula for a particular solution of the latter is derived, and the least of the orders of nonsingular matrices containing a given rectangular matrix as a submatrix is determined. 
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     author = {Yu. A. Alpin and S. N. Il'in},
     title = {The matrix equation $AX-YB=C$ and related problems},
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     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a1/}
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Yu. A. Alpin; S. N. Il'in. The matrix equation $AX-YB=C$ and related problems. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 15-23. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a1/