## Mathematics Faculty Articles

#### Title

On Taylor’s Formula for the Resolvent of a Complex Matrix

#### Document Type

Article

#### Publication Date

11-1-2008

#### Publication Title

Computers & Mathematics with Applications

#### Keywords

Powers of a matrix, Matrix invariants, Resolvent

#### ISSN

0898-1221

#### Volume

56

#### Issue/No.

9

#### First Page

2285

#### Last Page

2288

#### Abstract

The resolvent R_{λ}(A) of a complex r×r matrix A is an analytic function in any domain with empty intersection with the spectrum Σ_{A} of A. The well known Taylor expansion of R_{λ}(A) in a neighborhood of any given λ_{0}∉Σ_{A} is modified taking into account that only the first powers of R_{λ0}(A) are linearly independent. The main tool in this framework is given by the multivariable polynomials depending on the invariants v_{1},v_{2},…,v_{r} of R_{λ}(A) (m denotes the degree of the minimal polynomial). These functions are used in order to represent the coefficients of the subsequent powers of R_{λ0}(A) as a linear combination of the first m of them.

#### NSUWorks Citation

He, Matthew and Ricci, Paolo E., "On Taylor’s Formula for the Resolvent of a Complex Matrix" (2008). *Mathematics Faculty Articles*. 161.

http://nsuworks.nova.edu/math_facarticles/161

#### DOI

10.1016/j.camwa.2008.03.051