The Largest Entry in the Inverse of a Vandermonde Matrix

TL;DR

This paper studies the asymptotic behavior of the largest entry in the inverse of specific Vandermonde matrices, providing existence results and formulas for its limit as matrix size grows.

Contribution

It establishes the existence of the limit of the maximum entry in the inverse of certain Vandermonde matrices and derives formulas for this limit.

Findings

The limit of the maximum entry exists as matrix size tends to infinity.

Formulas for the limit of the maximum entry are provided.

The study applies to matrices with entries defined by powers of a real number greater than one.

Abstract

We investigate the size of the largest entry (in absolute value) in the inverse of certain Vandermonde matrices. More precisely, for every real $b > 1$, let $M_b(n)$ be the maximum of the absolute values of the entries of the inverse of the $n \times n$ matrix $[b^{i j}]_{0 \leq i, j < n}$. We prove that $\lim_{n \to +\infty} M_b(n)$ exists, and we provide some formulas for it.