SUMS OF IDEMPOTENTS

Let L(k) denote the set of those n X n matrices expressible as a sum o f k idempotent matrices. We study conditions for membership in L(k) wi th ''small'' k. It is shown that the nontrivial cases are those in whi ch the trace t of a matrix A does not exceed 2rho - 2, where rho is th e rank of A. For A to belong to L(k) it is sufficient that t (which is necessarily an integer at least equal to rho) be greater than 2rho 1 - k. In certain cases the results are shown to be sharp. For cyclic matrices and, more generally, for those with a low number of blocks in their rational canonical forms, improved results are obtained. Since the number of idempotent summands is often large, the problem of appro ximating a real or complex matrix by a member of L(k) is also consider ed. It is shown, for example, that if the trace of A is an integer t w ith rho less-than-or-equal-to t less-than-or-equal-to n, then A is in the closure of L3, while the smallest k with A is-an-element-of L(k) m ay be n.