Exponential polar factorization of the fundamental matrix of linear differential systems

Abstract

We propose a new constructive procedure to factorize the fundamental real matrix of a linear system of differential equations as the product of the exponentials of a symmetric and a skew-symmetric matrix. Both matrices are explicitly constructed as series whose terms are computed recursively. The procedure is shown to converge for sufficiently small times. In this way, explicit exponential representations for the factors in the analytic polar decomposition are found. An additional advantage of the algorithm proposed here is that, if the exact solution evolves in a certain Lie group, then it provides approximations that also belong to the same Lie group, thus preserving important qualitative properties.