On Taylors formula for the resolvent of a complex matrix
The resolvent R?(A) R ? ( A ) of a complex rr r  r matrix A A is an analytic function in any domain with empty intersection with the spectrum ?A ? A of A A . The well known Taylor expansion of R?(A) R ? ( A ) in a neighborhood of any given ?0??A ? 0 ? ? A is modified taking into account that only the first powers of R?0(A) R ? 0 ( A ) are linearly independent. The main tool in this framework is given by the multivariable polynomials F k , n ( v 1 , v 2 ,  , v r ) ( n = ? 1 , 0 , 1 ,  ; k = 1 , 2 ,  , m ? r ) depending on the invariants v1,v2,,vr v 1 , v 2 ,  , v r of R?(A) R ? ( A ) (m m denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of R?0(A) R ? 0 ( A ) as a linear combination of the first m m of them.
resolvent powers of a matrix matrix invariants