Document Open Access Logo

Generating All Invertible Matrices by Row Operations

Authors Petr Gregor , Hung P. Hoang , Arturo Merino , Ondřej Mička

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2024.35.pdf
  • Filesize: 0.87 MB
  • 14 pages

Document Identifiers

Author Details

Petr Gregor
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Hung P. Hoang
  • Algorithm and Complexity Group, Faculty of Informatics, TU Wien, Austria
Arturo Merino
  • Institute of Engineering Sciences, Universidad de O'Higgins, Rancagua, Chile
Ondřej Mička
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic

Funding

This work was supported by Czech Science Foundation grant GA 22-15272S.
  • Hoang, Hung P.: Austrian Science Foundation (FWF, project Y1329 START-Programm)
  • Merino, Arturo: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).

Acknowledgements

This work was initiated at the 2nd Combinatorics, Algorithms, and Geometry workshop in Dresden, Germany in 2022. We would like to thank the organizers and participants for the inspiring atmosphere.

Cite As Get BibTex

Petr Gregor, Hung P. Hoang, Arturo Merino, and Ondřej Mička. Generating All Invertible Matrices by Row Operations. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.35

Abstract

We show that all invertible n × n matrices over any finite field 𝔽_q can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or dividing a row by the generator of the multiplicative group of 𝔽_q. This even holds in the more general setting where the pairs of rows that can be added or subtracted are specified by an arbitrary transition tree that has to satisfy some mild constraints. Moreover, we can prescribe the first and the last matrix if n ≥ 3, or n = 2 and q > 2. In other words, the corresponding flip graph on all invertible n × n matrices over 𝔽_q is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Hamilton cycle
  • combinatorial Gray code
  • invertible matrices
  • finite field
  • general linear group
  • generation algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. J. Bang-Jensen and G. Gutin. Alternating cycles and paths in edge-coloured multigraphs: a survey. Discrete Math., 165/166:39-60, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00160-4.
  2. J. Bang-Jensen, G. Gutin, and A. Yeo. Properly coloured Hamiltonian paths in edge-coloured complete graphs. Discrete Appl. Math., 82(1-3):247-250, 1998. URL: https://doi.org/10.1016/S0166-218X(97)00062-0.
  3. M. Bánkfalvi and Zs. Bánkfalvi. Alternating Hamiltonian circuit in two-coloured complete graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 11-18. Academic Press, New York-London, 1968. Google Scholar
  4. C. C. Chen and N. F. Quimpo. On strongly Hamiltonian abelian group graphs. In Combinatorial mathematics, VIII (Geelong, 1980), volume 884 of Lecture Notes in Math., pages 23-34. Springer, Berlin-New York, 1981. URL: https://doi.org/10.1007/BFb0091805.
  5. C. Cooper. On the rank of random matrices. Random Structures Algorithms, 16(2):209-232, 2000. URL: https://doi.org/10.1002/(SICI)1098-2418(200003)16:2<209::AID-RSA6>3.0.CO;2-1.
  6. M. Dupuis and S. Wagon. Laceable knights. Ars Math. Contemp., 9:115-124, 2015. URL: https://doi.org/10.26493/1855-3974.420.3C5.
  7. T. Fabšič, O. Grošek, K. Nemoga, and P. Zajac. On generating invertible circulant binary matrices with a prescribed number of ones. Cryptogr. Commun., 10(1):159-175, 2018. URL: https://doi.org/10.1007/s12095-017-0239-4.
  8. P. Gregor, H. P. Hoang, A. Merino, and O. Mička. Generating all invertible matrices by row operations. arXiv preprint, 2024. Full preprint version of the present article. URL: https://arxiv.org/abs/2405.01863.
  9. P. Gregor, A. Merino, and T. Mütze. The Hamilton Compression of Highly Symmetric Graphs. In S. Szeider, R. Ganian, and A. Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 54:1-54:14, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.54.
  10. E. Hartung, H. P. Hoang, T. Mütze, and A. Williams. Combinatorial generation via permutation languages. I. Fundamentals. Trans. Amer. Math. Soc., 375(4):2255-2291, 2022. URL: https://doi.org/10.1090/tran/8199.
  11. D. E. Knuth. The Art of Computer Programming. Vol. 4A. Combinatorial algorithms. Part 1. Addison-Wesley, Upper Saddle River, NJ, 2011. Google Scholar
  12. L. Lovász. Problem 11. In Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf., Calgary, AB, 1969). Gordon and Breach, New York, 1970. Google Scholar
  13. T. Mütze. Combinatorial Gray codes - an updated survey. Electron. J. Combin., Dynamic Surveys DS26:93 pp., 2023. URL: https://doi.org/10.37236/11023.
  14. Naomi Nishimura. Reasons to fall (more) in love with combinatorial reconfiguration. In WALCOM: algorithms and computation, volume 14549 of Lecture Notes in Comput. Sci., pages 9-14. Springer, Singapore, 2024. URL: https://doi.org/10.1007/978-981-97-0566-5_2.
  15. David J. Rasmussen and Carla D. Savage. Hamilton-connected derangement graphs on S_n. Discrete Math., 133(1-3):217-223, 1994. URL: https://doi.org/10.1016/0012-365X(94)90028-0.
  16. Frank Ruskey and Carla Savage. Hamilton cycles that extend transposition matchings in Cayley graphs of S_n. SIAM J. Discrete Math., 6(1):152-166, 1993. URL: https://doi.org/10.1137/0406012.
  17. J. Sawada and A. Williams. Solving the sigma-tau problem. ACM Trans. Algorithms, 16(1):Art. 11, 17 pp., 2020. URL: https://doi.org/10.1145/3359589.
  18. M. Tchuente. Generation of permutations by graphical exchanges. Ars Combin., 14:115-122, 1982. Google Scholar
  19. W. C. Waterhouse. Two generators for the general linear groups over finite fields. Linear and Multilinear Algebra, 24(4):227-230, 1989. URL: https://doi.org/10.1080/03081088908817916.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact