  In this document we study the local connectivity of the sets whose elements
are $m$-tuples of pairwise commuting normal matrix contractions. Given
$\varepsilon>0$, we prove that there is $\delta>0$ such that for any two
$m$-tuples of pairwise commuting normal matrix contractions
$\mathbf{X}:=(X_1,\ldots,X_m)$ and
$\tilde{\mathbf{X}}:=(\tilde{X}_1,\ldots,\tilde{X}_m)$ that are $\delta$-close
with respect to some suitable distance $\eth$ in $(\mathbb{C}^{n\times n})^m$,
we can find a $m$-tuple of matrix paths (homotopies) connecting $\mathbf{X}$ to
$\mathbf{\tilde{X}}$ relative to the intersection of some
$\varepsilon,\eth$-neighborhood of $\mathbf{X}$ with the set of $m$-tuples of
pairwise commuting normal matrix contractions. One of the key features of these
matrix homotopies is that $\delta$ can be chosen independent of $n$.
  Some connections with topology and numerical matrix analysis will be outlined
as well.
