# On variation of eigenvalues of birth and death matrices and random walk matrices

## TL;DR

This paper improves known results on how the extreme eigenvalues of birth and death matrices and random walk matrices vary, and advances towards solving a long-standing open problem about their eigenvalue variation.

## Contribution

It provides enhanced bounds on eigenvalue variation and makes progress on a thirty-year-old open problem in the spectral analysis of these matrices.

## Key findings

- Improved bounds on eigenvalue variation for birth and death matrices.
- Progress towards solving the open problem on eigenvalue variation.
- Enhanced understanding of eigenvalue monotonicity in these matrices.

## Abstract

The purpose of this note is twofold: firstly to improve the known results on variation of extreme eigenvalues of birth and death matrices and random walk matrices; and secondly to progress towards the solution of a thirty years old open problem concerning the variation of eigenvalues of these matrices. Keywords: Birth and death matrices, random walk matrices, eigenvalues, monotonicity