# Perturbation bounds for the Mostow and the bipolar decompositions

## TL;DR

This paper derives perturbation bounds for Mostow's and bipolar matrix decompositions, utilizing new expressions for the derivative of the geometric mean of positive definite matrices.

## Contribution

It introduces new perturbation bounds for specific matrix decompositions and derives expressions for the derivative of the geometric mean of positive definite matrices.

## Key findings

- Perturbation bounds for Mostow's decomposition.
- Perturbation bounds for bipolar decomposition.
- Derived expressions for the derivative of the geometric mean.

## Abstract

Perturbation bounds for Mostow's decomposition and the bipolar decomposition of matrices have been computed. To do so, expressions for the derivative of the geometric mean of two positive definite matrices have been derived.