Sums of two triangularizable quadratic matrices over an arbitrary field (Q417457)

Let \(\mathbb{K}\) be an arbitrary field. Given a pair \((a,b)\in \mathbb{K}^{2}\), a matrix \(A\) of \(M_{n}(\mathbb{K})\) is called \((a,b)\)-quadratic when \(A^{2}=aA+bI_{n}\). Let \((a,b,c,d)\in \mathbb{K}^{4}\). A matrix is called an \((a,b,c,d)\)-quadratic sum when it can be decomposed as the sum of an \((a,b)\)-quadratic matrix and of a \((c,d)\)-quadratic one.NEWLINENEWLINEIn this paper, the author tries to give necessary and sufficient conditions for a matrix of \(M_{n}(\mathbb{K})\) to be an \((a,b,c,d)\)-quadratic sum. The case where \(b=d=0\), \(a\neq 0\) and \(c\neq 0\) has been dealt by \textit{C. de Seguins Pazzis} [ibid. 433, No. 3, 625--636 (2010; Zbl 1206.15014)] and the case \(a=b=c=d=0\) has been dealt by \textit{J. D. Botha} [ibid. 436, No. 3, 516--524 (2012; Zbl 1298.15026)]. The author studies the remaining case \(a=1\) and \(b=c=d=0\) in order to complete the case where both polynomial \(t^{2}-at-b\) and \(t^{2}-ct-d\) are split over \(\mathbb{K}\). Then, the author generalizes the results of \textit{J.-H. Wang} [ibid. 229, 127--149 (1995; Zbl 0833.15011)] to an arbitrary field, regardless of the characteristic of \(\mathbb{K}\), by determining which matrices can be decomposed as the sum of an idempotent and a square-zero matrix.