 
Summary: Inversion of analytic matrix functions
that are singular at the origin \Lambda
Konstantin E. Avrachenkov y Moshe Haviv z Phil G. Howlett x
August 7, 2000
Abstract
In this paper we study the inversion of an analytic matrix valued function A(z). This
problem can also be viewed as an analytic perturbation of the matrix A 0 = A(0). We
are mainly interested in the case where A 0 is singular but A(z) has an inverse in some
punctured disc around z = 0. It is known that A \Gamma1 (z) can be expanded as a Laurent
series at the origin. The main purpose of this paper is to provide eOEcient computational
procedures for the coeOEcients of this series. We demonstrate that the proposed algorithms
are computationally superior to symbolic algebra when the order of the pole is small.
Key words. matrix inversion, matrix valued functions, analytic perturbation, Lau
rent series.
AMS subject classiøcations. 15A09, 41A58, 47A55, 47A56
Abbreviated title. Inversion of analytic matrix functions.
1 Introduction
Let fA k g k=0;1;::: ` R n\Thetan be a sequence of matrices that deønes the analytic matrix valued
function
A(z) = A 0 + zA 1 + z 2 A 2 + \Delta \Delta \Delta : (1)
