On the Exponentials of Some Structured Matrices

TL;DR

This paper presents explicit algorithms and closed-form formulas for computing the exponentials of certain structured 4x4 matrices, utilizing Clifford algebra and Lie theory, with extensions to higher dimensions through structure-preserving transformations.

Contribution

It introduces new algorithms and formulas for matrix exponentials of structured matrices, leveraging Clifford algebra and Lie theory, and extends applicability beyond four dimensions.

Findings

Explicit algorithms for 4x4 structured matrices

Closed-form formulas for matrix exponentials

Extension to higher dimensions using structure-preserving similarities

Abstract

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.