The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices
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 / nullitySource:
Mathematics, 2020, 8, 12, 2147-Publisher:
- Basel : MDPI
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Institution/Community
JakovTY - JOUR AU - Tošić, Marina AU - Ljajko, Eugen AU - Kontrec, Nataša AU - Stojanović, Vladica PY - 2020 UR - https://jakov.kpu.edu.rs/handle/123456789/1635 AB - 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. PB - Basel : MDPI T2 - Mathematics T1 - The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices VL - 8 IS - 12 SP - 2147 DO - 10.3390/math8122147 ER -
@article{ author = "Tošić, Marina and Ljajko, Eugen and Kontrec, Nataša and Stojanović, Vladica", year = "2020", 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.", publisher = "Basel : MDPI", journal = "Mathematics", title = "The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices", volume = "8", number = "12", pages = "2147", doi = "10.3390/math8122147" }
Tošić, M., Ljajko, E., Kontrec, N.,& Stojanović, V.. (2020). The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices. in Mathematics Basel : MDPI., 8(12), 2147. https://doi.org/10.3390/math8122147
Tošić M, Ljajko E, Kontrec N, Stojanović V. The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices. in Mathematics. 2020;8(12):2147. doi:10.3390/math8122147 .
Tošić, Marina, Ljajko, Eugen, Kontrec, Nataša, Stojanović, Vladica, "The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices" in Mathematics, 8, no. 12 (2020):2147, https://doi.org/10.3390/math8122147 . .