The length problem for a sum of idempotents (Q1347222)

\textit{R. E. Hartwig} and \textit{M. S. Putcha} [Linear Multilinear Algebra 26, 279-286 (1990; Zbl 0696.15011)] and \textit{P. Y. Wu} [Linear Algebra Appl. 142, 43-54 (1990; Zbl 0724.15012)] independently proved that an \(n\times n\) complex matrix is a sum of idempotent matrices if and only if its trace is an integer \(\geq\) its rank. For positive integers \(n\), \(t\), let \(E_ n(t)\) denote the set of those \(n\times n\) matrices of rank \(t\) that are sums of idempotents, and let \(e_ n(t)\) denote the smallest \(m\) such that every member of \(E_ n(t)\) is a sum of \(m\) idempotents (reviewer's notation). In this interesting and ingenious paper, the author goes a long way towards solving the length problem, which is to determine \(e_ n(t)\) for all \(n\), \(t\). The complete solution is given in each of the following cases: (a) \(n\leq 5\); (b) \(t< n\); (c) \(t= mn\), where \(1\leq m\leq n- 2\); (d) \(t\geq n(n- 1)\). The values of \(t\) that escape (b)--(d) are those lying in one of the intervals \(I_ m\): \(mn< t< (m+ 1)n\), where \(1\leq m\leq n- 2\). For each such \(I_ m\), the author determines the maximum of \(e_ n(t)\) for \(t\in I_ m\). (Correction: in Theorem 2.9, \(n+ 1< \text{tr }T\) should be \(n+ 1\leq \text{tr }T\)).