A linear autonomous system of differential equations $\dot{x}=Ax$ can be transformed to its Jordan normal form, that is, the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. This result is generalized to arbitrary nonautonomous linear systems $\dot{x}=A(t)x$ with a locally integrable matrix function $A:{\bb R}\longrightarrow {\bb R}^{N\times N}$.