In this paper we present necessary and sufficient conditions for (full) output feedback equivalence of linear control systems. Given two minimal control systems with m inputs n states and r outputs, we give as solution the equality of certain invariants and the existence of solution of a linear system of n(m+r)-mr equations with a number of unknowns depending on the distribution of controllability and observability indices. The conditions degenerate to a complete set of invariants when the controllability indices are equal to each other and the observability indices like-wise, with particular case the scalar problem. The conditions do not allow in a direct way the parameterization of the output feedback orbits and they do not involve redundant data.