ar X iv : m at h - ph / 0 50 80 18 v 1 8 A ug 2 00 5 On The Exponential of Matrices in su ( 4 )

This note provides explicit techniques to compute the exponentials of a variety of anti-Hermitian matrices in dimension four. Many of these formulae can be written down directly from the entries of the matrix. Whenever any spectral calculations are required, these can be done in closed form. In many instances only 2 × 2 spectral calculations are required. These formulae cover a wide variety of applications. Conditions on the matrix which render it to admit one of three minimal polynomials are also given. Matrices with these minimal polyno-mials admit simple and tractable representations for their exponentials. One of these is the Euler-Rodrigues formula. The key technique is the relation between real 4 × 4 matrices and the quaternions.