the state-transition matrix
Engineers who model machines, circuits, and spacecraft as systems of differential equations need a single object that answers one question: given the state of the system now, what will the state be later? The state-transition matrix is exactly that operator. Multiply your current state vector by it and out comes the state at a future time — it transitions the state from one moment to another.
For a constant-coefficient system x' = A x, the state-transition matrix from time t0 to time t is e^(A(t - t0)), and if you start at t0 = 0 it is simply e^(At). It carries x(t0) to x(t) by x(t) = e^(A(t - t0)) x(t0). It has the natural properties you would demand of such a thing: transitioning from a time to itself does nothing (it equals I when t = t0), transitioning forward then forward again composes (the semigroup property, e^(A(t - s)) e^(A(s - t0)) = e^(A(t - t0))), and it can be inverted to run backward. For the general time-varying system x' = A(t) x, where no matrix exponential exists, the state-transition matrix is written Phi(t, t0) and is defined as the principal fundamental matrix that equals I at t0 — the genuinely general object.
The reason this term exists alongside 'matrix exponential' is mostly cultural: it is the language of control theory and systems engineering, where it underlies how a system's response is computed, how controllability and observability are defined, and how the full input-driven solution x(t) = Phi(t, t0) x(t0) + integral of Phi(t, s) g(s) ds is assembled. For constant A it is the same object as e^(At); for time-varying A it is the correct generalization that the bare exponential cannot provide.
A satellite's attitude obeys x' = A x near a reference orientation. Ground control measures the state x(t0) at one pass and wants the state x(t1) at the next. They compute the state-transition matrix Phi = e^(A(t1 - t0)) once, then every prediction is just one matrix-vector product x(t1) = Phi x(t0). The same Phi, reused, also propagates the uncertainty in the estimate forward in time.
Same object as e^(At) for constant A, but the name and Phi(t, t0) notation generalize to time-varying systems where no exponential exists.
For time-varying A(t) you cannot just write e^(integral of A) — that fails unless A(t) commutes with itself at different times; the state-transition matrix Phi(t, t0) is the correct object, defined by Phi' = A(t) Phi, Phi(t0, t0) = I, not by a naive exponential.