the matrix exponential by diagonalization
The series definition of e^(At) is honest but unfriendly to a calculator — you would be summing infinitely many matrix powers. Diagonalization is the trick that makes the computation almost free whenever the matrix is nice. The idea: change to a coordinate system in which the system decouples into separate one-variable problems, solve each trivially, then change back.
Here is the method in plain steps. Find the eigenvalues lambda_1, ..., lambda_n of A and a full set of eigenvectors. Stack the eigenvectors as the columns of a matrix P, and put the eigenvalues down the diagonal of a matrix D. Then A = P D P^(-1) — that is what diagonalizable means. The point is that powers collapse beautifully: A^k = P D^k P^(-1), and D^k is just each diagonal entry raised to the k-th power. Feed this into the exponential series and the P and P^(-1) factor out of every term, leaving e^(At) = P e^(Dt) P^(-1). And e^(Dt) is trivial: a diagonal matrix with e^(lambda_1 t), ..., e^(lambda_n t) on the diagonal. So the recipe is: build P and D, exponentiate the diagonal scalar by scalar, and sandwich with P and P^(-1).
This works exactly when A has a full set of independent eigenvectors — guaranteed if the eigenvalues are all distinct, and often (not always) when they repeat. If an eigenvalue is short of eigenvectors, A cannot be diagonalized and this clean formula stops; you fall back to generalized eigenvectors and the Jordan form. When eigenvalues are complex, the formula still holds with complex entries, and pairing conjugates afterward recovers real sines and cosines. Diagonalization is the workhorse first method — reach for it before anything fancier.
Let A = [1, 1; 0, 2]. Eigenvalues are 1 and 2, with eigenvectors (1, 0) and (1, 1). So P = [1, 1; 0, 1], D = [1, 0; 0, 2], and P^(-1) = [1, -1; 0, 1]. Then e^(At) = P [e^t, 0; 0, e^(2t)] P^(-1) = [e^t, e^(2t) - e^t; 0, e^(2t)]. Check: at t = 0 this is I, and differentiating gives A e^(At). No infinite series was ever summed by hand.
Diagonalization moves to eigen-coordinates where each mode grows as a pure e^(lambda t), then transforms the answer back.
Diagonalization needs a full set of independent eigenvectors; a defective matrix (too few eigenvectors for a repeated eigenvalue) cannot be diagonalized, and the P e^(Dt) P^(-1) formula simply does not apply.