  In this work we examine the interplay between normal form and matched
particle distributions in a linear setting. We first outline the connection
between the established sigma-matrix method and Williamson's Theorem. Then we
show that the Iwasawa decomposition provides a natural framework for a
description of beam optics parameters. Along the way we will apply these
methods to a realistic tracking example, as well as provide additional
examples, including the connection to the parameterization of Courant-Synder.
