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On Linear Dependence of Rows and Columns in Matrices over Non-commutative Domains

Published: 05 July 2022 Publication History

Abstract

Some well-known correspondences between sets of linearly independent rows and columns of matrices over fields carry over to matrices over non-commutative rings without nontrivial zero divisors.

References

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N. Buaphim, K. Onsaard, P. So-ngoen, and T. Rungratgasame. Some reviews on ranks of upper triangular block matrices over a skew field. International Mathematical Forum, 13(7):323--335, 2018.
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P.M. Cohn. On the embedding of rings in skew fields. Proc. London Math. Soc., 11:511--530, 1961.
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P.M. Cohn. Fields of fractions. Amer. Math. Monthly, 78(6):596--617, 1971.
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P.M. Cohn. Free rings and their relations. Academic Press, London, New York, 1971.
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D.K. Faddeev. Lectures in algebra (Russian). "Nauka", Moscow, 1984.
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D. Knuth. The art of computer programming. Vol. 4A / Combinatorial algorithms, volume 1. Addison-Wesley, New York, 2014.
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J.B. Lopatinsky. Theory of general boundary problems (Russian). Naukova Dumka. Kyev, 1964.
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Maple Help. Ore algebra: annihilators, 2021.
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O. Ore. Linear equations in non-commutative fields. Ann. Math., 32:463--477, 1931.
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B.L. Van der Waerden. Algebra, volume 1. Springer New York, 1970.
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J.H.M. Wedderburn. Non-commutative domains of integrity. J. Reine Angew. Math., 167:129--141, 1932.

Cited By

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  • (2023)On Ranks of Matrices over Noncommutative DomainsЖурнал вычислительной математики и математической физики10.31857/S004446692305002263:5(760-762)Online publication date: 1-May-2023
  • (2023)On Ranks of Matrices over Noncommutative DomainsComputational Mathematics and Mathematical Physics10.1134/S096554252305002063:5(771-778)Online publication date: 27-Jun-2023

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  1. On Linear Dependence of Rows and Columns in Matrices over Non-commutative Domains

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    cover image ACM Conferences
    ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
    July 2022
    547 pages
    ISBN:9781450386883
    DOI:10.1145/3476446
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    Publication History

    Published: 05 July 2022

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    Author Tags

    1. linear dependence of rows (columns)
    2. matrices over domains
    3. maximum sets of linearly independent rows (columns)
    4. non-commutative domains
    5. quotient skew fields
    6. row and column ranks

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    • (2023)On Ranks of Matrices over Noncommutative DomainsЖурнал вычислительной математики и математической физики10.31857/S004446692305002263:5(760-762)Online publication date: 1-May-2023
    • (2023)On Ranks of Matrices over Noncommutative DomainsComputational Mathematics and Mathematical Physics10.1134/S096554252305002063:5(771-778)Online publication date: 27-Jun-2023

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