On Uniform Connectivity of Algebraic Matrix Sets

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 $ð$ 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 $δ$-ball $B_ð(\mathbf{X},δ)$ centered at $\mathbf{X}$ for some $δ>0$, with respect to $ð$. Two of the key features of these matrix paths is that $δ$ can be chosen independent of $n$, and that they are contained in the intersection of $B_ð(\mathbf{X},\varepsilon)$ 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.