The left greatest common divisor and the left least common multiple of all solutions of the matrix equation BX = A over a commutative elementary divisor domain
Abstract
UDC 512.64
The explicit form of the left greatest common divisor and the left least common multiple of all solutions to the solvable matrix equation $BX=A$ over a commutative domain of elementary divisors is given.
It is proved that they are solutions too.
References
W. J. Vetter, Vector structures and solutions of linear matrix equations, Linear Algebra and Appl., 10, № 2, 181 – 188 (1975), https://doi.org/10.1016/0024-3795(75)90010-5 DOI: https://doi.org/10.1016/0024-3795(75)90010-5
J. R. Magnus, H. Neudecker, The elimination matrix: some lemmas and applications, SIAM J. Algebraic and Discrete Methods, 1, № 4, 422 – 449 (1980), https://doi.org/10.1137/0601049 DOI: https://doi.org/10.1137/0601049
F. J. Don, On the symmetric solutions of a linear matrix equation, Linear Algebra and Appl., 93, 1 – 7 (1987), https://doi.org/10.1016/S0024-3795(87)90308-9 DOI: https://doi.org/10.1016/S0024-3795(87)90308-9
C. G. Khatri, S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, J. Appl. Math., 31, № 4, 579 – 585 (1976), https://doi.org/10.1137/0131050 DOI: https://doi.org/10.1137/0131050
A. Ran, M. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc.Amer. Math. Soc., 132, № 5, 1435 – 1443 (2004), https://doi.org/10.1090/S0002-9939-03-07220-4 DOI: https://doi.org/10.1090/S0002-9939-03-07220-4
B. Recht, M. Fazel, P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, https://doi.org/10.1137/070697835. DOI: https://doi.org/10.1137/070697835
J. R. Magnus, $L$-structured matrices and linear matrix equations, Linear Algebra and Appl., 14, № 1, 67 – 88 (1983), https://doi.org/10.1080/03081088308817543 DOI: https://doi.org/10.1080/03081088308817543
I. Kaplansky, Elementary divisor and modules, Trans. Amer. Math. Soc., 66, 464 – 491 (1949), https://doi.org/10.2307/1990591 DOI: https://doi.org/10.2307/1990591
V. A. Bovdi, V. P. Shchedryk, Commutative Bezout domains of stable range 1.5, Linear Algebra and Appl., 568, 127 – 134 (2019), https://doi.org/10.1016/j.laa.2018.06.012 DOI: https://doi.org/10.1016/j.laa.2018.06.012
B. V. Zabavsky, Diagonal reduction of matrices over rings, Math. Stud. Monogr. Ser. (2012).
P. S. Kazimirs`kij, Rozklad matrichnikh mnogochleniv na mnozhniki, Nauk. dumka, Kiyiv (1981).
. V. P. Shchedryk, Faktorizacziya matricz` nad kil`czyami elementarnikh dil`nikiv, L`viv: In-t prikl. probl. mekhaniki i matematiki NAN Ukrayini, L`viv (2017).
V. P. Shchedryk, Factorization of matrices over elementary divisor domain, Algebra and Discrete Math., № 2, 79 – 99 (2009).
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