On Taylor's formula for the resolvent of a complex matrix
The resolvent R-lambda(A) of a complex r x r matrix A is an analytic function in any domain with empty intersection with the spectrum Sigma(A) of A. The well known Taylor expansion of R-lambda(A) in a neighborhood of any given lambda(0) is not an element of Sigma(A) is modified taking into account that only the first powers of R-lambda 0 (A) are linearly independent. The main tool in this framework is given by the multivariable polynomials F-k,F-n(v(1), v(2),...,v(r)) (n = -1, 0, 1,...; k = 1, 2,...,m <= r) depending on the invariants v(1), v(2),...,v(r) of R-lambda(A) (in denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of R-lambda 0 (A) as a linear combination of the first m of them. (C) 2008 Elsevier Ltd. All rights reserved.
resolvent powers of a matrix matrix invariants