Sums of two triangularizable quadratic matrices over an arbitrary field

Details Sums of two triangularizable quadratic matrices over an arbitrary field Sums of two triangularizable quadratic matrices over an arbitrary field

2011-09-10

arxiv.org/abs/1109.2255v4

Mathematics

Rings and Algebras

14 pages

Scientific paper

10.1016/j.laa.2011.11.026

Let K be an arbitrary field, and a,b,c,d be elements of K such that the polynomials t^2-at-b and t^2-ct-d are split in K[t]. Given a square matrix M with entries in K, we give necessary and sufficient conditions for the existence of two matrices A and B such that M=A+B, A^2=a A+bI_n and B^2=c B+dI_n. Prior to this paper, such conditions were known in the case b=d=0, a<>0 and c<>0, and also in the case a=b=c=d=0. Here, we complete the study, which essentially amounts to determining when a matrix is the sum of an idempotent and a square-zero matrix. This generalizes results of Wang to an arbitrary field, possibly of characteristic 2.

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