Every matrix is a linear combination of three idempotents
It is shown that every matrix \(A\in \mathbb{F}^{n\times n}\) over a field \(\mathbb{F}\) of characteristic zero may be represented as \(\lambda_1 Q_1+ \lambda_2 Q_2+ \lambda_3 Q_2\), where \(\lambda_i\in \mathbb{F}\), \(Q_i\in \mathbb{F}^{n\times n}\) and \(Q^2_i= Q_i\). If \(A\in \mathbb{F}^{2\times 2}\) or \(A\in \mathbb{C}^{3\times 3}\) it is fulfilled \(A= \lambda_1 Q_1+ \lambda_2 Q_2\). At the same time there are matrices \(A\in \mathbb{Q}^{3\times 3}\) and \(A\in \mathbb{C}^{4\times 4}\) that are not linear combinations of two idempotents.
scientific article; zbMATH DE number 2116528