On Taylor’s formula for the resolvent of a complex matrix

The resolvent of a complex r×r matrix is an analytic function in any domain with empty intersection with the spectrum of . The well known Taylor expansion of in a neighborhood of any given is modified taking into account that only the first powers of are linearly independent. The main tool in this framework is given by the multivariable polynomials depending on the invariants v1,v2,…,vr of (m denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of as a linear combination of the first m of them.