Matrix exponentials, SU(N) group elements, and real polynomial roots

TL;DR

This paper explores expressing SU(N) group elements as matrix polynomials with coefficients involving elementary functions and invariants related to eigenvalues, linking geometric simplex projections to matrix eigenvalues.

Contribution

It provides a novel polynomial representation of SU(N) elements using geometric invariants and eigenvalue projections, connecting algebraic and geometric perspectives.

Findings

Matrix exponential expressed as polynomial of order N-1

SU(N) elements represented via polynomial with trigonometric coefficients

Eigenvalues linked to simplex vertex projections and geometric invariants

Abstract

The exponential of an NxN matrix can always be expressed as a matrix polynomial of order N-1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N-1 in a traceless NxN hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N-2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N-vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an (N-1)-simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.