matrix exponential via Jordan form

The matrix exponential exp(A) = I + A + A^2/2! + ... is the key to linear differential equations: the solution of x'(t) = A x(t) with x(0) = x_0 is x(t) = exp(t A) x_0. The Jordan form turns this infinite series into a short, exact, block-by-block computation.

Write A = P J P^-1. Then exp(t A) = P exp(t J) P^-1, and exp(t J) is block-diagonal with one exponential per Jordan block. For a single block, exp(t J_k(lambda)) = e^{t lambda} * exp(t N), where N is the nilpotent shift; since N^k = 0, exp(t N) = I + t N + t^2 N^2/2! + ... + t^{k-1} N^{k-1}/(k-1)! is a finite polynomial in t.

So the entries of exp(t A) are combinations of e^{lambda t} times polynomials in t, with the polynomial degree governed by the block size: a size-k block contributes terms up to t^{k-1} e^{lambda t}. This is exactly why solutions of a defective linear ODE system carry t-power factors (resonance) rather than pure exponentials.

This resurfaces well beyond ODEs: in the matrix-groups track, exp maps a Lie algebra to a Lie group, and one-parameter subgroups t -> exp(t A) are the basic flows. The Jordan picture of exp explains their growth and oscillation. (For a complex lambda = a + i b, e^{lambda t} = e^{a t}(cos b t + i sin b t) splits growth from rotation.)