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Maximal and Minimal Triangular Matrices

  • Roksana Słowik
Open Access
Article

Abstract

We call a nonscalar matrix maximal (or minimal) if its centralizer is maximal (respectively minimal) in the poset of all centralizers of matrices. We discuss the form of maximal and minimal matrices in the algebra of upper triangular matrices.

Keywords

Centralizer triangular matrices 

Mathematics Subject Classification

15A21 13E10 

References

  1. 1.
    Aiat Hadj Ahmed, D., Słowik, R.: m-commuting maps of the rings of infinite triangular and strictly triangular matrices. (in preparation)Google Scholar
  2. 2.
    Arenas-Herrera, M.I., Verde-Star, L.: Representation of doubly infinite matrices as non-commutative Laurent series. Special Matrices 5, 250–257 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bhatia, R., Rosenthal, P.: How and why to solve the operator equation \(AX-XB=Y\). Bull. Lond. Math. Soc. 29, 1–21 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brown, W.C.: Constructing maximal commutative subalgebra of matrix rings in small dimensions. Commun. Algebra 25, 3923–3946 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dolinar, G., Guterman, A., Kuzma, B., Oblak, P.: Commuting graphs and extremal centralizers. Ars Math. Contemp. 7, 453–459 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dolinar, G., Guterman, A., Kuzma, B., Oblak, P.: Extremal matrix centralizers. Linear Algebra Appl. 438, 2904–2910 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Drazin, M.: Some generalizations of matrix commutativity. Proc. Lond. Math. Soc. 3–1, 222–231 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drazin, M.P., Dungey, J.W., Gruenberg, K.W.: Some theorems on commutative matrices. J. Lond. Math. Soc. 26, 221–228 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frobenius, G.: Über lineare Substitution und Bilinear Formen, Vol. 84 , pp. 1–63. Grelle (1878)Google Scholar
  10. 10.
    Gerstenhaber, M.: On dominance and varieties of commuting matrices. Ann. Math. 73, 324–348 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guterman, A.E., Markova, O.V.: Commutative matrix subalgebras and length function. Linear Algebra Appl. 430, 1790–1805 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoffman, K., Kunze, R.: Linear Algebra, 2nd edn. Prentice-Hall In, Engelwood Cllifs (1971)zbMATHGoogle Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lagerstrom, P.: A proof of a theorem on commutative matrices. Bull. Am. Math. Soc. 51, 535–536 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Šemrl, P.: Non-linear commutativity preserving maps. Acta Sci. Math. (Szeged) 71, 781–819 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Słowik, R.: Maps on infinite triangular matrices preserving idempotents. Linear Multilinear Algebra 62, 938–964 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Słowik, R.: Every infinite triangular matrix is similar to a generalized infinite Jordan matrix. Linear Multilinear Algebra 65, 1362–1373 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Słowik, R.: Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix (Linear Multilinear Algebra 65, 1362–1373 (2017))’. Linear Multilinear Algebra.  https://doi.org/10.1080/03081087.2017.1397094
  19. 19.
    Thijsse, P.: Upper triangular similarity of upper triangular similarity of upper triangular matrices. Linear Algebra Appl. 260, 119–149 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, S.: The Jordan normal form of infinite matrices. Chin. Sci. Bull. 41, 1943–1946 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wedderburn, J.H.M.: Lectures on Matrices. Dover Publications Inc, New York (1964)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsSilesian University of TechnologyGliwicePoland

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