the Jordan form method
/ zhor-DAHN (French) /
Diagonalization is the dream, but sometimes a matrix is stubborn: an eigenvalue repeats, yet it does not supply enough independent eigenvectors to fill out the basis. Such a matrix is called defective, and the clean formula e^(At) = P e^(Dt) P^(-1) has nowhere to land. The Jordan form is the next-best canonical shape — as close to diagonal as a defective matrix can get — and it tells you exactly how to compute e^(At) anyway.
Instead of a purely diagonal D, you bring A to a form J that is diagonal except for a few 1's sitting just above the diagonal, grouped into blocks. Each Jordan block has a single eigenvalue lambda down its diagonal and 1's on the superdiagonal. The trick that makes the exponential tractable is to split the block as lambda I + N, where N is the part with the 1's. N is nilpotent — some power of it is the zero matrix — so e^(Nt) is not an infinite series at all but a short polynomial in t (I + Nt + N^2 t^2/2! + ... stopping the moment N's powers die). Then, because lambda I commutes with N, the block's exponential is e^(lambda t) times that finite polynomial. This is precisely where the extra factors of t in the solution come from: a 2-by-2 block produces solutions like e^(lambda t) and t e^(lambda t), echoing the repeated-root case of a single equation. You build e^(At) by exponentiating each block this way and conjugating by the change-of-basis matrix of generalized eigenvectors.
In practice you rarely form the full Jordan structure for hand computation; for a single repeated eigenvalue you can short-circuit straight to the answer using generalized eigenvectors or the Cayley-Hamilton-based methods. The Jordan form's real value is conceptual: it proves that e^(At) always exists and always has the shape (polynomial in t) times (exponentials), and it explains the t, t^2, ... factors as a precise consequence of missing eigenvectors.
Take A = [3, 1; 0, 3] — one eigenvalue 3 but only one eigenvector, so it is defective. It is already a single Jordan block: write A = 3I + N with N = [0, 1; 0, 0], and N^2 = 0. Then e^(Nt) = I + Nt = [1, t; 0, 1], and e^(At) = e^(3t) [1, t; 0, 1] = [e^(3t), t e^(3t); 0, e^(3t)]. The factor t e^(3t) is the fingerprint of the missing eigenvector.
A Jordan block splits into lambda I plus a nilpotent N, and N's exponential is a finite polynomial in t — the source of every extra t factor.
You only need the Jordan form when the matrix is defective; if eigenvectors are plentiful, diagonalization is simpler, and forming an exact Jordan form numerically is notoriously unstable, so it is mostly a theoretical and hand-computation tool.