In 2003, Jung, Ko and Pearcy used the Aluthge transform of an n-by-n matrix to build a unitarily invariant, norm-decreasing retraction of the space of all n-by-n matrices onto the space of normal n-by-n matrices.  We show how to join their analysis to Geometric Invariant Theory in order to identify the C*-envelopes of certain algebras generated by d-fold tuples of n-by-n matrices together with their traces.