So here's what it tells you. It says that to get this state at time Tau plus T from the state at time Tau – first of all they're linearly related. That alone is – well, it's not unexpected, but they're linearly related. They're related by an end-by-end matrix that maps one to the other. And that matrix is simply E to the TA. So E to the TA is a time propagator. It propagates X dot equals AX forward T seconds in time. If T is negative it actually runs time backwards, and reconstructs what the state was some seconds ago. ...

See the whole transcript at Introduction to Linear Dynamical Systems - Lecture 12