The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices
1
Department of Mathematics, Faculty of Sciences, University in Priština, 38220 Kosovska Mitrovica, Serbia
2
Department of Informatics, University of Criminal Investigation and Police Studies, 11000 Belgrade, Serbia
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2147; https://doi.org/10.3390/math8122147
Received: 25 October 2020
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Revised: 18 November 2020
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Accepted: 19 November 2020
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Published: 2 December 2020
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.
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Keywords:
k-potent matrix; linear combination; nonsingularity; rank; nullity
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MDPI and ACS Style
Tošić, M.; Ljajko, E.; Kontrec, N.; Stojanović, V. The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices. Mathematics 2020, 8, 2147. https://doi.org/10.3390/math8122147
AMA Style
Tošić M, Ljajko E, Kontrec N, Stojanović V. The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices. Mathematics. 2020; 8(12):2147. https://doi.org/10.3390/math8122147
Chicago/Turabian StyleTošić, Marina, Eugen Ljajko, Nataša Kontrec, and Vladica Stojanović. 2020. "The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices" Mathematics 8, no. 12: 2147. https://doi.org/10.3390/math8122147
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