The exponential map of GL(N)

TL;DR

This paper presents a finite, explicit expansion for the matrix exponential of an N×N matrix, utilizing the Cayley-Hamilton theorem to express higher powers and resulting in rational functions of eigenvalues.

Contribution

It introduces a finite expansion of the matrix exponential for N×N matrices using Cayley-Hamilton theorem, simplifying calculations with explicit formulas.

Findings

Finite expansion of matrix exponential derived

Expresses higher powers as linear combinations of first N-1 powers

Results are rational functions of eigenvalues

Abstract

A finite expansion of the exponential map for a $N\times N$ matrix is presented. The method uses the Cayley-Hamilton theorem for writing the higher matrix powers in terms of the first N-1 ones. The resulting sums over the corresponding coefficients are rational functions of the eigenvalues of the matrix.