Sums of two triangularizable quadratic matrices over an arbitrary field

TL;DR

This paper characterizes when a matrix over an arbitrary field can be expressed as the sum of two matrices, each satisfying specific quadratic polynomial equations, extending previous results to fields of any characteristic.

Contribution

It provides necessary and sufficient conditions for such matrix decompositions over any field, generalizing earlier work to include fields of characteristic 2.

Findings

Conditions for matrix sums with quadratic polynomial constraints are established.

The results extend to arbitrary fields, including characteristic 2.

The study completes the characterization of matrices as sums of an idempotent and a square-zero matrix.

Abstract

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.