Local Deformation of Matrix Words

In this document we study some local deformation properties of matrix representations of the universal C$^*$-algebras denoted by $\mathbb{I}^{m}_\varepsilon[p_1,\ldots,p_m]$ and $\mathbb{S}^{m-1}_\varepsilon[p_1,\ldots,p_m]$, and that we call {\bf Semi-Soft Cubes} and {\bf Semi-Soft Spheres} respectively. We will use some C$^*$-algebraic technology to study the local deformation properties of matrix words in particular representations of Semi-Soft Cubes and Spheres, we will then use these results to study the local deformation properties of generic matrix equations on words. Some geometrical aspects of the local deformation of matrix words will be addressed, and some connections with matrix numerical analysis and computational physics will be outlined as well.