On the Exponential of Matrices in su(4)

TL;DR

This paper derives explicit formulas for exponentiating 4x4 anti-Hermitian matrices using quaternion algebra, providing verifiable conditions for minimal polynomials and illustrating applications.

Contribution

It introduces explicit exponential formulas for 4x4 anti-Hermitian matrices and characterizes conditions for their minimal polynomials using quaternion algebra.

Findings

Explicit exponential formulas for matrices in su(4)

Conditions for minimal polynomials of these matrices

Applications demonstrating the formulas' utility

Abstract

This note presents explicit formulae for the exponentials of a wide variety of matrices which are 4x4, anti-Hermitian. Easily verifiable conditions characterizing when such matrices admit one of three minimal polynomials are also given. Essential use of the algebra isomorphism between real 4x4 matrices and the tensor product of the quaternions with themseleves is made. Illustrations from important applications are given.