  We show constructively that, under certain regularity assumptions, any system
of coupled linear differential equations with variable coefficients can be
tridiagonalized by a time-dependent Lanczos-like method. The proof we present
formally establishes the convergence of the Lanczos-like algorithm and yields a
full characterization of algorithmic breakdowns. From there, the solution of
the original differential system is available in closed form. This is a key
piece in evaluating the elusive ordered exponential function both formally and
numerically.
