Jordan normal form via ODE's

Abstract

In text books on differential equations the system of ordinary differential equations with constant coefficients X' = AX is often solved by reduction (by an invertible change of variables X = PY) to the simpler system Y' = JY where J is the Jordan canonical form of A. Here we do things the other way around and deduce the existence of J and P by comparing two types of solutions of the system X' = AX. The proof provides a straight-forward algorithm for calculating the matrices J and P above. Apart from some elementary considerations on (formal) solutions of systems of ODE's with constant coefficients, the main ingredient of the proof (and of the resulting algorithm) is one which comes up in other approaches, namely the reduction of polynomial matrices (in one variable) to diagonal form by row and column operations.