Matrix analysis and omega calculus.

Abstract

In this work we introduce a new operator based approach to matrix analysis. Our main technical tool comprises an extension of a tool introduced long ago by MacMahon to analyze the partitions of natural numbers: the Omega operator calculus. More precisely, we construct an operator acting linearly on absolutely convergent matrix valued expansions which selects appropriate terms of those expansions. In the context of our framework a new representation of matrix valued functions is available. Our representation is simple, requiring only the computation of matrix inverses and basic manipulations of the Taylor series of scalar functions. To show the usefulness of our approach we obtain fundamental results related to the basic theory of ODEs, perturbative calculations, multiple integrals involving the matrix exponential, the Sylvester equation, the multivariate Fa`a di Bruno formula, Hermite polynomials, queuing theory, and graph theory.