Sums of two triangularizable quadratic matrices over an arbitrary field

Let K be an arbitrary field, and a,b,c,d be elements of K such that the polynomials t2-at-b and t2-ct-d are split in K[t]. Given a square matrix M∈Mn(K), we give necessary and sufficient conditions for the existence of two matrices A and B such that M=A+B, A2=aA+bIn and B2=cB+dIn. Prior to this paper, such conditions were known in the case b=d=0, and [4], and in the case a=b=c=d=0 [1], . 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 [5] to an arbitrary field, possibly of characteristic 2.