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

Marina Tošić, Eugen Ljajko, Nataša Kontrec and Vladica Stojanović
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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

Mathematics, 2020, vol. 8, issue 12, 1-13

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.

Keywords: k-potent matrix; linear combination; nonsingularity; rank; nullity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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