# On the smallest singular value of multivariate Vandermonde matrices with clustered nodes

## TL;DR

This paper establishes bounds on the smallest singular value of multivariate Vandermonde matrices with clustered nodes on the complex unit circle, revealing how clustering affects matrix stability and invertibility.

## Contribution

It provides the first comprehensive bounds for the smallest singular value in multivariate Vandermonde matrices with clustered nodes, including sharp constants and geometric dependencies.

## Key findings

- Lower bounds for singular values with clustered nodes
- Upper bounds that match the univariate case
- Dependence of singular value on cluster geometry

## Abstract

We prove lower bounds for the smallest singular value of rectangular, multivariate Vandermonde matrices with nodes on the complex unit circle. The nodes are ``off the grid'', groups of nodes cluster, and the studied minimal singular value is bounded below by the product of inverted distances of a node to all other nodes in the specific cluster. By providing also upper bounds for the smallest singular value, this completely settles the univariate case and pairs of nodes in the multivariate case, both including reasonable sharp constants. For larger clusters, we show that the smallest singular value depends also on the geometric configuration within a cluster.