# Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

## TL;DR

This paper investigates conditions under which a matrix function with an unstable set of partial indices can be approximated by another function with the same partial indices, facilitating factorization despite instability.

## Contribution

It introduces a method to construct approximate matrix functions with stable partial indices, addressing challenges posed by unstable index sets.

## Key findings

- Conditions for constructing approximate functions with stable partial indices
- Method for close approximation of original matrix functions
- Potential to improve factorization techniques for unstable cases

## Abstract

From the classic work of Gohberg and Krein (1958), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the larger and the smaller partial indices is larger than unity then, in any neighborhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when imposed on the original matrix function, allow to construct another matrix function that exhibits the same partial indices and is close to the original matrix function.