Dynamical Deformation of Toroidal Matrix Varieties

TL;DR

This paper proves local connectivity for sets of commuting normal matrix contractions, showing that close matrices can be connected via paths that are independent of matrix size, with implications for topology and numerical analysis.

Contribution

The work establishes size-independent bounds for homotopies connecting close commuting normal matrix contractions, advancing understanding of their topological structure.

Findings

Homotopies exist between close commuting normal matrices.

Size-independent bounds for matrix paths are proven.

Connections to topology and numerical analysis are discussed.

Abstract

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.