Dynamical Deformation of Toroidal Matrix Varieties

Abstract

In this document we study the local connectivity of the sets whose elements are $m$-tuples of pairwise commuting normal matrix contractions. Given $\epsilon >0$, we prove that there is $\delta >0$ such that for any two $m$-tuples of pairwise commuting normal matrix contractions $\mathbf{X}:=(X_1,\dots ,X_m)$ and $\tilde{\mathbf{X}}:=({\tilde{X}}_1,\dots ,{\tilde{X}}_m)$ that are $\delta$-close with respect to some suitable distance $ð$ in $({\mathbb{C}}^{n\times n}{)}^m$, we can find a $m$-tuple of matrix paths (homotopies) connecting $\mathbf{X}$ to $\tilde{\mathbf{X}}$ relative to the intersection of some $\epsilon ,ð$-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.

References in corpus (3)

• K-Theory and Pseudospectra for Topological Insulators
• Local Matrix Homotopies and Soft Tori
• Local Deformation of Matrix Words