Products of simultaneously triangulable idempotent matrices (Q755835)

Let A be a square matrix over an arbitrary field. In order that A is the product of simultaneously triangulable idempotents it is necessary and sufficient that 0 and 1 are the only possible eigenvalues of A and that the order of 1 as zero of the minimal polynomial of A does not exceed the order of 0 as zero of the characteristic polynomial of A by more than one. This result is deduced from the following proposition: If R denotes the ring of upper triangular matrices, then \(A=(a_{ij})^ n_{i,j=1}\in R\) is the product of idempotents in R if and only if \(a_{ii}=0\) or 1 for all i and \(a_{ii}=1\), \(k\leq i\leq \ell\) implies that \((a_{ij})^{\ell}_{i,j=k}\) is the \(\ell +1-k\)-identity matrix.