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This article is cited in 1 scientific paper (total in 1 paper)
Ordinary differential equations
On ranks of matrices over noncommutative domains
S. A. Abramova, M. Petkovšekb, A. A. Ryabenkoa a Federal Research Center "Computer Science and Control" of the Russian Academy of Science, 119333 Moscow, Vavilova str., 40, Russia
b University of Ljubljana, Faculty of Mathematics and Physics
SI-1000 Ljubljana, Jadranska 19, Slovenia
Abstract:
We consider matrices with entries in some domain $R$, 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 $R$; 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 $R$ is a necessary and sufficient condition for the equality of the row and column ranks of an arbitrary matrix over $R$. An algorithm for calculating the rank of a given matrix is proposed. Our Maple implementation of this algorithm covers the domains of differential and ($q$-)difference operators, both ordinary and with partial derivatives and differences.
Received: 30.08.2022 Revised: 30.09.2022 Accepted: 02.02.2023
Citation:
S. A. Abramov, M. Petkovšek, A. A. Ryabenko, “On ranks of matrices over noncommutative domains”, Zh. Vychisl. Mat. Mat. Fiz., 63:5 (2023), 760–762; Comput. Math. Math. Phys., 63:5 (2023), 771–778
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https://www.mathnet.ru/eng/zvmmf11551 https://www.mathnet.ru/eng/zvmmf/v63/i5/p760
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