On some special matrix decompositions over fields and finite commutative rings

Danchev, Peter ; Garc´ıa, Esther ; Lozano, Miguel G´omez

Mathematics and Education in Mathematics, Tome 50 (2021), pp. 95-101

Résumé

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 square-zero nilpotent matrix. In addition, each 2 × 2 matrix over any field admits such a representation. We also 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. As a consequence of these decompositions, we show that every matrix over a finite field can be expressed as the sum of a potent matrix and a square-zero matrix. Moreover, we prove that every matrix over a finite commutative ring is always representable as a sum of a potent matrix and a square-zero nilpotent matrix, provided the Jacobson radical of the ring has zero-square.Our main theorems substantially improve on recent results due to Abyzov et al. in Mat. Zametki (2017), Ster in Lin. Algebra & Appl. (2018), Breaz in Lin. Algebra & ˇAppl. (2018) and Shitov in Indag. Math. (2019).

Détail

Keywords: Companion matrix, Jordan normal form, rational form, irreducible polynomial, field, finite commutative ring, nilpotent matrix, potent matrix, 15A24, 15B33, 16U99