The Exponential Matrix: an explicit formula by an elementary method

TL;DR

This paper presents a simple, elementary formula for computing the exponential of a real square matrix, avoiding complex methods like Jordan forms or eigenvector calculations, using only power series and partial fractions.

Contribution

It introduces an explicit formula for the matrix exponential that simplifies calculations and proofs, requiring minimal advanced tools.

Findings

Provides an explicit formula for $e^{tA}$

Demonstrates the method with two examples

Proves a classical stability result

Abstract

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $n\times n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor resolution of linear systems of differential equations, nor resolution of linear systems with constant coefficients, nor matrix inversion, nor complex integration, nor functional analysis. The basic tools are power series and the method of partial fraction decomposition. Two examples are given. A proof of one well-known stability result is given.