We consider minimality under external equivalence for linear, time-invariant, implicit (E, A, B, C) descriptions of the type Ex(t) = Ax(t) + Bu(t), y(t) = Cx(t). Necessary and sufficient conditions for external minimality have been previously given, from an algebraic point of view by Kuijper. We propose here alternative geometric characterizations, as well as a geometric reduction procedure that extracts, from a given (E, A, B, C) description, a minimal model of the same type. We also geometrically characterize, in terms of the geometry of (E, A, B, C), the corresponding minimal dimensions for the state space and for the state equation space.