On Uniform Connectivity of Algebraic Matrix Sets

TL;DR

This paper establishes uniform local path connectivity for sets of commuting normal matrices constrained by polynomial equations, with paths that are independent of matrix size and contained within specified neighborhoods.

Contribution

It proves the existence of size-independent matrix paths within constrained matrix sets, advancing understanding of their connectivity properties.

Findings

Paths exist between matrices within polynomial constraint sets.

Paths can be chosen independently of matrix size n.

Paths are contained within specified neighborhoods.

Abstract

In this document we study the uniform local path connectivity of sets of $m$-tuples of pairwise commuting normal matrices with some additional constraints. More specifically, given given $\varepsilon>0$, a fixed metric $\eth$ in ${M_n(\mathbb{C})}^m$ induced by the operator norm $\|\cdot\|$, any collection of $r$ non-constant multivariable polynomials $p_1(x_1,\ldots,x_m),\ldots,p_r(x_1,\ldots,x_m)$ over $\mathbb{C}$ with finite zero set $\mathbf{Z}(p_1,\ldots,p_r)\subset \mathbb{C}^m$, and any $m$-tuple $\mathbf{X}=(X_1,\ldots,X_m)$ in the set $\mathbb{ZD}_n^m(p_1,\ldots,p_r)\subseteq M_n^m(\mathbb{C})$, of pairwise commuting normal matrix contractions such that, $\|p_j(Y_1,\ldots,Y_m)\|=0$ for each $(Y_1,\ldots,Y_m)\in \mathbb{ZD}_n^m(p_1,\ldots,p_r)$ and each $1\leq j\leq r$. We prove the existence of paths between arbitrary $m$-tuples, that lie in the intersection of $\mathbb{ZD}_n^m(p_1,\ldots,p_r)$, and the $\delta$-ball $B_\eth(\mathbf{X},\delta)$ centered at $\mathbf{X}$ for some $\delta>0$, with respect to $\eth$. Two of the key features of these matrix paths is that $\delta$ can be chosen independent of $n$, and that they are contained in the intersection of $B_\eth(\mathbf{X},\varepsilon)$ and $\mathbb{ZD}_n^m(p_1,\ldots,p_r)$. Some connections with the approximation theory for matrix functions of several matrix variables, are studied as well.