Decompositions of matrices into diagonalizable and square-zero matrices

Abstract

For any n ≥ 2 and fixed k ≥ 1, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring Mn(F) to be written as a sum of an invertible matrix U and a nilpotent matrix N with N k = 0 over an arbitrary field F.

On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

“…The construction is based on Jordan blocks and roots of unity. Let us consider the explicit decomposition for a Jordan block (see Remarks 2.8 and 2.9 from [8]).…”

“…Since K is algebraically closed and A is similar to a direct sum of Jordan blocks then it is sufficient to decompose each Jordan block. Let J be a Jordan block of size m × m for some m n and q is the characteristic of K. If m 2 the decomposition is straightforward (see Section 1 of [8]). When m 3 and if q does no divide m then decomposition of J is presented in Lemma 2.1(i).…”

“…Let us consider below two situations, namely, algebraically closed fields (see Corollary 2.4) and finite fields (see Corollary 3.2). The results can be viewed and treated as the development of method and ideas presented in [8] and [6], respectively. Some closely related studies can also be found in [5].…”

Certain Properties of Square Matrices over Fields with Applications to Rings

“…A brief collection of principally known historical facts in this branch are as follows: In [6] was shown that any square matrix over the finite two elements field Z 2 is a sum of a nilpotent matrix and an idempotent matrix; thereby the full matrix n × n ring M n (Z 2 ) is called nil-clean. This important fact was strengthened in [23] by showing that, for any n ∈ N and for every n × n matrix A over Z 2 , there exists an idempotent matrix E such that (A − E) 4 = 0, while over the finite indecomposable ring Z 4 consisting of four elements this relation is precisely (A − E) 8 = 0 (see [2] and [22] for some further generalizations and specifications, too). In [23] is showed also that the ring ∞ n=1 M n (Z 2 ) is both nil-clean and von Neumann regular but not strongly π-regular, whereas the ring ∞ n=1 M n (Z 4 ) is both nil-clean and regularly nil clean but not π-regular (see [9], as well).…”

“…The aim of this short article is to settle this conjecture in the case of algebraically closed fields. This will be successfully done in the sequel (compare with [12] and [8] as well). Further eventual applications of such decompositions could be realized in coding theory and, in particular, in noncommutative coding theory (cf.…”