We introduce the notion of a generalized attractor-repeller pair for ẋ = A(t)x which in fact is an attractor-repeller pair for the induced (nonlinear) flow on the projective space, and use it to characterize diagonalizability. We show that integral separation implies the existence of generalized attractor-repeller pairs, and use topological dynamics, more precisely the hull of A(t), to prove a converse theorem. The relation to exponential separation is also discussed.