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 n with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators. Given the n × n coefficient matrix A(t), one wishes to solve the initial-value problem associated with the linear ordinary differential equation for the unknown n-dimensional vector function Y(t). When n = 1, the solution simply reads 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 A is independent of t. 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): which is subsequently constructed as a series expansion: where, for simplicity, it is customary to write Ω(t) for Ω(t, t0) and to take t0 = 0. Magnus appreciated that, since d/dt (eΩ) e−Ω = A(t), using a Poincaré−Hausdorff matrix identity, he could relate the time derivative of Ω to the generating function of Bernoulli numbers and the adjoint endomorphism of Ω, to solve for Ω recursively in terms of A "in a continuous analog of the BCH 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 where [A, B] ≡ A B − B A is the matrix commutator of A and B. These equations may be interpreted as follows: Ω1(t) coincides exactly with the exponent in the scalar (n = 1) case, but this equation cannot give the whole solution.