Abstract
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 square nonsingular matrices containing a given rectangular matrix as a submatrix is determined. Bibliography: 5 titles.
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W. E. Roth, “The equations AX-Y B = C and AX-XB = C in matrices,” Proc. Amer. Math. Soc., 3, 392–396 (1953).
H. Flanders and H. K. Wimmer, “On matrix equations AX-XB = C and AX-YB = C,” SIAM J. Appl. Math., 32, 707–710 (1977).
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F. R. Gantmakher, The Theory of Matrices [in Russian], Nauka, Moscow (1967).
V. N. Malozemov, M. F. Monako, and A. A. Petrov, “Formulas of Frobenius, Sherman-Morrison, and related issues,” Zh. Vychisl. Matem. Matem. Fiz., 42, No. 10, 1459–1465 (2002).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 15–23.
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Al’pin, Y.A., Il’in, S.N. The matrix equation AX-YB=C and related problems. J Math Sci 137, 4769–4773 (2006). https://doi.org/10.1007/s10958-006-0273-9
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DOI: https://doi.org/10.1007/s10958-006-0273-9