the Cayley-Hamilton theorem
/ KAY-lee HAM-il-tun /
Every square matrix has a characteristic polynomial — the polynomial in lambda whose roots are the eigenvalues, found from det(A - lambda I) = 0. The Cayley-Hamilton theorem makes a startling claim: if you take that polynomial and plug the matrix A in for the variable lambda, you get the zero matrix. In short, every matrix satisfies its own characteristic equation. The eigenvalues are the lambda's that kill the polynomial as a number; the theorem says A itself kills it as a matrix.
Concretely, if A is n-by-n with characteristic polynomial p(lambda) = lambda^n + c_(n-1) lambda^(n-1) + ... + c_1 lambda + c_0, then p(A) = A^n + c_(n-1) A^(n-1) + ... + c_1 A + c_0 I = 0. The immediate payoff for differential equations is enormous. This equation lets you rewrite A^n in terms of lower powers I, A, A^2, ..., A^(n-1) — and then A^(n+1), A^(n+2), and every higher power too, by repeated substitution. So the infinite exponential series, which seems to need all powers of A, actually collapses: e^(At) can always be written as a finite combination c_0(t) I + c_1(t) A + ... + c_(n-1)(t) A^(n-1), with only n scalar coefficient functions to determine. The boundless series is secretly finite-dimensional.
This is the engine behind two practical methods. One finds the coefficient functions by demanding that the same finite formula reproduce e^(lambda t) for each eigenvalue lambda (handling repeats by also matching derivatives). The other, Putzer's algorithm, generates them recursively without ever computing eigenvectors. Either way, Cayley-Hamilton is the reason you never need an infinite sum: an n-dimensional system's exponential lives in the span of just I, A, ..., A^(n-1).
For A = [0, 1; -2, -3], the characteristic polynomial is lambda^2 + 3 lambda + 2. Cayley-Hamilton says A^2 + 3A + 2I = 0, i.e. A^2 = -3A - 2I — you can check this by squaring A directly. Because every power of A reduces to a combination of I and A, you write e^(At) = c_0(t) I + c_1(t) A and find c_0, c_1 by matching e^(-t) and e^(-2t) at the eigenvalues -1 and -2.
Because A obeys its own degree-n equation, A^n and all higher powers fold back into I, A, ..., A^(n-1) — so the exponential series is secretly finite.
Cayley-Hamilton is about substituting the matrix into the polynomial — substitute A for lambda and you get the zero matrix, not the zero number; do not confuse it with the eigenvalue equation det(A - lambda I) = 0, which is solved by numbers.