Powers via Diagonalization

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Computing A^ n via Diagonalization Use A = PDP^ -1 to raise a matrix to a high power without 100 multiplications. Let . Computing A^ 10 by direct multiplication means multiplying A by itself nine times. Instead, diagonalize A and use A^ n = PD^ n P^ -1 to find A^ 10 — and a formula for A^ n — almost instantly. Nice work — you turned ten matrix multiplications into one power of a diagonal matrix. The payoff: The master formula: if A = PDP^ -1 , then A^ n = PD^ n P^ -1 , because every interior P^ -1 P collapses to the identity. Powers of a diagonal matrix are free: — just raise each eigenvalue to the n . No matrix multiplication. Cost is constant in n : A^ 10 , A^ 100 , or A^ 1000 all take the same three matrix operations once A is diagonalized. You get a closed form: here , exact for every n . The same idea powers Markov-chain long-run behavior, systems of linear differential equations, and the matrix exponential e^ At — all of which lean on raising a diagonalized matrix to a power.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.