On uniform connectivity of algebraic matrix sets

Abstract

We study the uniform path connectivity of sets of matrix tuples that satisfy some additional constraints, and more specifically, given $\epsilon >0$, a fixed metric $ð$ in $M_n(\mathbb{C}{)}^m$ induced by the operator norm $\Vert \cdot \Vert$, any collection of $r$ nonconstant polynomials $p_1(x_1,\dots ,x_m),\dots ,p_r(x_1,\dots ,x_m)$ over $\mathbb{C}$ with finite zero set $\mathbf{Z}(p_1,\dots ,p_r)\subset {\mathbb{C}}^m$ and any $m$-tuple $\mathbf{X}=(X_1,\dots ,X_m)$ in the set ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)\subseteq M_n^m\left(\mathbb{C}\right)$ of commuting normal matrix contractions such that $\Vert p_j(Y_1,\dots ,Y_m)\Vert =0$ for each $(Y_1,\dots ,Y_m)\in {\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$ and each $1\le j\le r$. The author proves the existence of paths between arbitrary $m$-tuples that belong to the intersection of ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$ and the open $\delta$-ball $B_{ð}(\mathbf{X},\delta )$ centered at $\mathbf{X}$ for some $\delta >0$ that can be chosen independently of $n$. In addition, the author proves that the aforementioned paths are contained in the intersection of $B_{ð}(\mathbf{X},\epsilon )$ and ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$. Some connections of the main results with structure-preserving perturbation theory and preconditioning techniques are outlined.