  We determine those k-tuples of conjugacy classes of matrices, from which it
is possible to choose matrices which have no common invariant subspace and have
sum zero. This is an additive version of the Deligne-Simpson problem. We deduce
the result from earlier work of ours on preprojective algebras and the moment
map for representations of quivers. Our answer depends on the root system for a
Kac-Moody Lie algebra.
