Diagonalization

Raise a matrix to the 1000th power by multiplying it out and you'll wait lifetimes. But if you rotate into the right basis — the eigen-basis — the matrix becomes diagonal, and diagonal matrices raise to any power by just exponentiating each diagonal entry. Diagonalization, A = PDP⁻¹, is the trick that turns exponential matrix computations into trivial ones, and it underlies Markov chains, systems of differential equations, and Google's PageRank.