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

The resolvent Rλ(A)Rλ(A) of a complex r×rr×r matrix AA is an analytic function in any domain with empty intersection with the spectrum ΣAΣA of AA. The well known Taylor expansion of Rλ(A)Rλ(A) in a neighborhood of any given λ0∉ΣAλ0∉ΣA is modified taking into account that only the first powers of Rλ0(A)Rλ0(A) are linearly independent. The main tool in this framework is given by the multivariable polynomials Fk,n(v1,v2,…,vr)(n=−1,0,1,…;k=1,2,…,m≤r) depending on the invariants v1,v2,…,vrv1,v2,…,vr of Rλ(A)Rλ(A) (mm denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of Rλ0(A)Rλ0(A) as a linear combination of the first mm of them.