Computational matrix representation modules for linear operators with explicit constructions for a class of lie operators

Abstract

A linear mapping from a finite-dimensional linear space to another has a matrix representation. Certain multilinear functions are also matrix-representable. Using these representations, symbolic computations can be done numerically and hence more efficiently. This paper presents an organized procedure for constructing matrix representations for a class of linear operators on finite-dimensional spaces. First we present serial number functions for locating basis monomials in the linear space of homogeneous polynomials of fixed degree, ordered under structured lexicographies. Next basic lemmas describing the modular structure of matrix representations for operators constructed canonically from elementary operators are presented. Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprised of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly. The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poission bracket.