[ndriscoll]

In addition to the differential equation, you can also tweak the definition of the exponential function as a limit used with e.g. compound interest:

    exp(tA) = lim n->infinity (I+tA/n)^n
    = lim n->infinity (I+tA/n)...(I+tA/n) (n times)

So you can interpret A (or log T) as a direction to move from the identity, and exp does infinite iterated compositions of an infinitesimal shift away from the identity in that direction.