Products of nilpotent matrices

In 1975, Paul Halmos asked: when can a square matrix be written as a product of nilpotent matrices? This was answered indirectly by Sourour in 1992: if n >= 3 then any n x n singular matrix can be written in this way. In this article, we determine the best possible index for the nilpotents in such a product, and compare our work on linear transformations of a vector space with analogous results for transformations of a set.