computing matrix powers via diagonalization
Multiplying a matrix by itself many times is expensive and unilluminating. But if A is diagonalizable, A = P D P^-1, then powers become almost free: A^k = P D^k P^-1, and raising a diagonal matrix to a power just raises each diagonal entry to that power. The hard repeated multiplication is replaced by exponentiating numbers.
The reason is telescoping. A^2 = (P D P^-1)(P D P^-1) = P D (P^-1 P) D P^-1 = P D^2 P^-1, and the same cancellation repeats k times to give A^k = P D^k P^-1. Diagonalizing once, then plugging in any k, yields a closed form valid for every power simultaneously — including, by the same idea, e^A = P e^D P^-1 and other matrix functions.
The signature application is solving linear recurrences in closed form. Write a recurrence as a vector update u_{k+1} = A u_k; then u_k = A^k u_0 = P D^k P^-1 u_0, and reading off the entries gives an explicit formula. For Fibonacci, the 2x2 companion matrix has eigenvalues the golden ratio phi and its conjugate, and this is exactly where Binet's formula F_k = (phi^k - psi^k)/sqrt(5) comes from.
Diagonalize once, then every power is obtained by exponentiating the diagonal eigenvalues.
When A is defective and cannot be diagonalized, the same program still runs with the Jordan form A = P J P^-1: powers J^k of a Jordan block have a tidy closed form involving binomial coefficients, so closed-form powers survive even without a full eigenbasis.