Solving the matrix exponential function for special orthogonal groups SO(n) and the exceptional G$_2$

TL;DR

This paper derives analytical formulas for the matrix exponential in special orthogonal groups up to dimension 9, using polynomial equations and invariants, and explores the exceptional Lie group G2 via these methods.

Contribution

It provides explicit analytical solutions for the matrix exponential in SO(n) for n up to 9, including special cases like G2, using polynomial equations and invariants.

Findings

Explicit formulas involve solving polynomial equations of degree up to four.

Matrix exponential trace expressed as sums of cosines of angles.

Construction of G2 via matrix exponential with a specific invariant constraint.

Abstract

In this work the matrix exponential function is solved analytically for the special orthogonal groups $SO(n)$ up to $n=9$. The number of occurring $k$-th matrix powers gets limited to $0\leq k \leq n-1$ by exploiting the Cayley-Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm $V$ and the roots of a polynomial equation that depends on a few specific invariants. Besides the well known case of $SO(3)$, a quadratic equation needs to be solved for $n=4,5$, a cubic equation for $n=6,7$, and a quartic equation for $n=8,9$. As an interesting subgroup of $SO(7)$, the exceptional Lie group $G_2$ of dimension $14$ is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, $\xi=1$. The calculation of the trace of the $SO(n)$-matrices arising from the exponential function, results in a sum of cosines of several angles, which specify the associated conjugation class as a point on a maximal torus.