On the Discretization of Double-Bracket Flows

Abstract.

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow Y'=[[Y,N],Y] , Y(0)=Y ₀ ∈\Sym(n) , can be represented in the form Y(t)=e ^Ω(t) Y ₀ e ^-Ω(t) , where the Taylor expansion of Ω can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolor leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite ``alphabet.''