Matrix Functions in SU (N ) F. Bruckmann, T. Heinzl, T. Tok and A. Wipf Friedrich-Schiller-Universit¨at Theoretisch-Physikalisches Institut Max-Wien-Platz 1 07743 Jena Germany Abstract We explicitly calculate the Riemannian metric on SU (2) and derive the associated Haar measure. For arbitrary elements of SU (N ) we derive a polar representation in terms of the eigenvalues and polynomials of the Lie algebra elements. The results are used to discuss the reducibility of gauge fields.