We study the uniform path connectivity of sets of matrix tuples that satisfy some additional constraints, and more specifically, given , a fixed metric in induced by the operator norm , any collection of nonconstant polynomials over with finite zero set and any -tuple in the set of commuting normal matrix contractions such that for each and each . The author proves the existence of paths between arbitrary -tuples that belong to the intersection of and the open -ball centered at for some that can be chosen independently of . In addition, the author proves that the aforementioned paths are contained in the intersection of and . Some connections of the main results with structure-preserving perturbation theory and preconditioning techniques are outlined.
"On uniform connectivity of algebraic matrix sets." Banach J. Math. Anal. 13 (4) 918 - 943, October 2019. https://doi.org/10.1215/17358787-2019-0009