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The Nullity, Rank, and Invertibility of Linear Combinations of k -Potent Matrices

Author

Listed:
  • Marina Tošić

    (Department of Mathematics, Faculty of Sciences, University in Priština, 38220 Kosovska Mitrovica, Serbia)

  • Eugen Ljajko

    (Department of Mathematics, Faculty of Sciences, University in Priština, 38220 Kosovska Mitrovica, Serbia)

  • Nataša Kontrec

    (Department of Mathematics, Faculty of Sciences, University in Priština, 38220 Kosovska Mitrovica, Serbia)

  • Vladica Stojanović

    (Department of Mathematics, Faculty of Sciences, University in Priština, 38220 Kosovska Mitrovica, Serbia
    Department of Informatics, University of Criminal Investigation and Police Studies, 11000 Belgrade, Serbia)

Abstract

Baksalary et al. (Linear Algebra Appl., doi:10.1016/j.laa.2004.02.025, 2004) investigated the invertibility of a linear combination of idempotent matrices. This result was improved by Koliha et al. (Linear Algebra Appl., doi:10.1016/j.laa.2006.01.011, 2006) by showing that the rank of a linear combination of two idempotents is constant. In this paper, we consider similar problems for k -potent matrices. We study the rank and the nullity of a linear combination of two commuting k -potent matrices. Furthermore, the problem of the nonsingularity of linear combinations of two or three k -potent matrices is considered under some conditions. In these situations, we derive explicit formulae of their inverses.

Suggested Citation

  • Marina Tošić & Eugen Ljajko & Nataša Kontrec & Vladica Stojanović, 2020. "The Nullity, Rank, and Invertibility of Linear Combinations of k -Potent Matrices," Mathematics, MDPI, vol. 8(12), pages 1-13, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2147-:d:454766
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