A Magnus expansion for the equation $Y'=AY-YB$

The subject matter of this paper is the representation of the solution of the linear differential equation $Y'=AY-YB$, $Y(0)=Y_0$, in the form $Y(t)=\ee^{\Omega(t)}Y_0$ and the representation of the function $\Omega$ as a generalisation of the classical Magnus expansion. An immediate application is a new recursive algorithm for the derivation of the Baker--Campbell--Hausdorff formula and its symmetric generalisation.