# Magnus expansion

In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first order homogeneous linear differential equation for a linear operator. In particular it furnishes the fundamental matrix of a system of linear ordinary differential equations of order Template:Mvar with varying coefficients. The exponent is built up as an infinite series whose terms involve multiple integrals and nested commutators.

## Magnus approach and its interpretation

Given the n × n coefficient matrix A(t), one wishes to solve the initial value problem associated with the linear ordinary differential equation

${\displaystyle Y^{\prime }(t)=A(t)Y(t),\qquad \qquad Y(t_{0})=Y_{0}}$

for the unknown Template:Mvar-dimensional vector function Y(t).

When n = 1, the solution simply reads

${\displaystyle Y(t)=\exp \left(\int _{t_{0}}^{t}A(s)\,ds\right)Y_{0}.}$

This is still valid for n > 1 if the matrix A(t) satisfies A(t1) A(t2) = A(t2) A(t1) for any pair of values of t, t1 and t2. In particular, this is the case if the matrix Template:Mvar is independent of Template:Mvar. In the general case, however, the expression above is no longer the solution of the problem.

The approach introduced by Magnus to solve the matrix initial value problem is to express the solution by means of the exponential of a certain n × n matrix function Ω(t, t0),

${\displaystyle Y(t)=\exp \left(\Omega (t,t_{0})\right)\,Y_{0}~,}$

which is subsequently constructed as a series expansion,

${\displaystyle \Omega (t)=\sum _{k=1}^{\infty }\Omega _{k}(t),}$

where, for simplicity, it is customary to write Ω(t) for Ω(t, t0) and to take t0 = 0.

Magnus appreciated that, since (ddt eΩ) e−Ω = A(t), using a Poincaré−Hausdorff matrix identity, he could relate the time-derivative of Template:Mvar to the generating function of Bernoulli numbers and the adjoint endomorphism of Template:Mvar,

${\displaystyle \Omega '={\frac {\operatorname {ad} _{\Omega }}{\exp(\operatorname {ad} _{\Omega })-1}}~A~,}$

to solve for Template:Mvar recursively in terms of Template:Mvar, "in a continuous analog of the CBH expansion", as outlined in a subsequent section.

The equation above constitutes the Magnus expansion or Magnus series for the solution of matrix linear initial value problem. The first four terms of this series read

${\displaystyle \int _{0}^{T}\|A(s)\|_{2}ds<\pi }$