On the computation of the nth power of a matrix

2017/05/14 by Nikolaos Halidias, Halidias, Nikolaos
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1705.04994

openalex publication_date 2017/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Evaluating matrix power series with the Cayley-Hamilton theorem
• Remarks on two theorems in linear algebra
• Efficient Computation of the Characteristic Polynomial
• Leveraging the Cayley Hamilton Theorem for Efficiently Solving the Jordan Canonical Form Problem
• A derivation of Dickson polynomials using the Cayley-Hamilton theorem