A note on sums of three square-zero matrices

2016/05/17 by Pazzis, Clément de Seguins
#15A24 #15B33 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1605.05131

Abstract

It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix A over an arbitrary field, there is a non-negative integer p such that the extended matrix A ⊕ 0_p is the sum of three square-zero matrices (more precisely, one can simply take p as the number of rows of A). Moreover, we demonstrate that if the underlying field has characteristic 2 then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the latter result for sums of idempotents.

Related

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• Sums and products of square-zero matrices
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• Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices
• Companion matrices as sums of p-potent and nilpotent matrices