On Ranks of Matrices over Noncommutative Domains
- Authors: Abramov S.A.1, Petkovšek M.2, Ryabenko A.A.1
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Affiliations:
- Federal Research Center “Computer Science and Control”, Russian Academy of Sciences
- Faculty of Mathematics and Physics, University of Ljubljana
- Issue: Vol 63, No 5 (2023)
- Pages: 760-762
- Section: ОБЫКНОВЕННЫЕ ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ
- URL: https://journals.rcsi.science/0044-4669/article/view/136132
- DOI: https://doi.org/10.31857/S0044466923050022
- EDN: https://elibrary.ru/DXCTUA
- ID: 136132
Cite item
Abstract
We consider matrices with entries in some domain, i.e., in a ring, not necessarily commutative, not containing non-trivial zero divisors. The concepts of the row rank and the column rank are discussed. (Coefficients of linear dependencies belong to the domain ; left coefficients are used for rows, right coefficients for columns.) Assuming that the domain satisfies the Ore conditions, i.e., the existence of non-zero left and right common multiples for arbitrary non-zero elements, it is proven that these row and column ranks are equal, which allows us to speak about the rank of a matrix without specifying which rank (row or column) is meant. In fact, the existence of non-zero left and right common multiples for arbitrary non-zero elements of is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and (-)difference operators, both ordinary and with partial derivatives and differences.
About the authors
S. A. Abramov
Federal Research Center “Computer Science and Control”, Russian Academy of Sciences
Email: sergeyabramov@mail.ru
Moscow, Russia
M. Petkovšek
Faculty of Mathematics and Physics, University of Ljubljana
Email: Marko.Petkovsek@fmf.uni-lj.si
Ljubljana, Slovenia
A. A. Ryabenko
Federal Research Center “Computer Science and Control”, Russian Academy of Sciences
Author for correspondence.
Email: anna.ryabenko@gmail.com
Moscow, Russia
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