# On the computation of the nth power of a matrix

## TL;DR

This paper explores methods for computing the nth power of a matrix, emphasizing the Cayley-Hamilton theorem and Gauss elimination to find the minimum polynomial, relevant to Markov chains and other areas.

## Contribution

It highlights the importance of the Cayley-Hamilton theorem and demonstrates Gauss elimination as a practical tool for matrix power computation.

## Key findings

- Cayley-Hamilton theorem is crucial for matrix power calculation.
- Gauss elimination can determine the minimum polynomial without the characteristic polynomial.
- The approach is applicable to Markov chains and related mathematical topics.

## Abstract

In this note we discuss the problem of finding the nth power of a matrix which is strongly connected to the study of Markov chains and others mathematical topics. We observe the known fact (but maybe not well known) that the Cayley-Hamilton theorem is of key importance to this goal. We also demonstrate the classical Gauss elimination technique as a tool to compute the minimum polynomial of a matrix without necessarily know the characteristic polynomial.