Decompositions of matrices into diagonalizable and square-zero matrices

ABSTRACT

In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a nilpotent matrix of order less than or equal to two. In addition, each 2 × 2 matrix over any field admits such a representation. We, moreover, show that, for all natural numbers n ≥  3, every n × n matrix over a finite field having no less than n + 1 elements also admits such a decomposition. The latter completes a recent example due to Breaz [Matrices over finite fields as sums of periodic and nilpotent elements. Linear Algebra Appl. 2018;555:92–97]. As a consequence of these decompositions, we show that every nilpotent matrix over a field can be expressed as the sum of a potent matrix and a square-zero matrix. This somewhat improves on recent results due to Abyzov et al. [On some matrix analogues of the little Fermat theorem. Mat Zametki. 2017;101(2):163–168] and Shitov [The ring ${\mathbb{M}}_{8k+4}\left({\mathbb{Z}}_2\right)$ is nil-clean of index four. Indag Math (N.S.). 2019;30:1077–1078].

Keywords:

• Companion matrix
• diagonalizable matrix
• nilpotent matrix
• Jordan normal form
• rational form
• irreducible polynomial

2010 Mathematics Subject Classifications:

• 15A24
• 15B33
• 16U99

Acknowledgments

The authors express their sincere thanks to the two anonymous expert referees for the careful reading of the manuscript and the competent insightful comments and suggestions made which improve substantially the structural shape of the presentation. The authors are also very grateful to the handling editor, Professor Stephane Gaubert, for his professional editorial management. The first named author is also very thankful to Professor Yaroslav Shitov for their valuable correspondence on the subject, which led to the main Question that motivated the writing of this paper, and on Remark 2.2.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the Bulgarian National Science Fund under Grant KP-06 N 32/1 of 7 December 2019. The second two authors were partially supported by MTM2017-84194-P ( AEI/FEDER, UE), and by the Junta de Andalucía FQM264.