  An example due to Pisier shows that two commuting, completely polynomially
bounded Hilbert space operators may not be simultaneously similar to
contractions. Thus, while each operator is individually similar to a
contraction, the pair is not jointly similar to a pair of commuting
contractions. We show that this phenomenon does not occur in finite dimensions.
More precisely, we show that a finite family of power bounded commuting
matrices is always jointly similar to a family of contractions. In fact, the
result can be extended to infinite families satisfying certain uniformity
conditions. Our approach is based on a joint spectral decomposition of the
underlying space.
