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

Abstract

The exponential of an N × N 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 N × N 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.