the matrix exponential
/ e to the A t /
Picture the simplest growth law there is: y' = a y, one number changing at a rate proportional to its own size. Its solution is y(t) = e^(at) y(0) — the constant a, fed into the exponential, tells the whole future from the starting value. Now imagine not one quantity but several, all tugging on one another at once: the temperatures of three connected rooms, the populations of interacting species, the currents in a circuit. The rule becomes x' = A x, where x is a column of numbers and A is a square matrix of rate constants. The matrix exponential is the answer to the obvious question: can we still just write x(t) = e^(At) x(0)? Remarkably, yes.
The definition copies the scalar exponential exactly. For a number, e^x = 1 + x + x^2/2! + x^3/3! + .... For a square matrix A you replace the number by the matrix and the 1 by the identity matrix I: e^(At) = I + At + (At)^2/2! + (At)^3/3! + .... Each term is a matrix (powers of a matrix are again matrices), and the infinite sum always converges, for every square matrix and every t. The result e^(At) is itself a square matrix, the same size as A. Two facts make it the right object: its derivative with respect to t is A e^(At) (differentiate the series term by term), and at t = 0 it equals I. So x(t) = e^(At) x(0) satisfies x' = A x and starts at x(0) — it is the solution of the initial value problem.
What you must not do is compute e^(At) entry by entry as if each slot of the matrix got exponentiated — that is wrong, because matrix multiplication mixes the rows and columns. The honest practical content of this field is that the series is a clean definition but a poor calculator; in real life you find e^(At) through eigenvalues and diagonalization, through the Jordan form when eigenvectors run short, through the Cayley-Hamilton theorem and Putzer's algorithm, or through the Laplace transform. The series is the meaning; those are the methods.
Let A = [0, 1; -1, 0], the matrix of the system x' = y, y' = -x (the harmonic oscillator). Computing the series, the even powers of At cycle through I and -I while the odd powers cycle through the rotation pieces, and the two sub-series are exactly the Taylor series of cosine and sine. The sum is e^(At) = [cos t, sin t; -sin t, cos t] — a rotation by angle t. So the matrix exponential of this A literally turns the plane, which is why its solutions go in circles.
e^(At) need not look exponential — for a rotation-type A it comes out as sines and cosines, because complex eigenvalues hide trig inside the exponential.
Never exponentiate a matrix entry by entry — e^(At) is defined by the power series, not by applying e^x to each slot, and the two almost never agree.