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arXiv · 2604.15135

On the exponential rate of the condition number of Fourier submatrices and Vandermonde matrices

Rikhav Shah·John Urschel
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Abstract

The discrete Fourier transform matrix is one of the most important matrices in linear algebra, and submatrices of it arise in a variety of applications. Though the discrete Fourier transform matrix is unitary, its submatrices can be exponentially ill-conditioned, an obstacle to accurate computation. This work resolves the exact rate of the exponential ill-conditioning for square submatrices with contiguous rows and columns. As a consequence, we obtain a tight upper bound of $2 G/\pi$ on the exponential rate for all submatrices with contiguous columns, or, equivalently, all Vandermonde submatrices with distinct support points, where $G$ is Catalan's constant. These results follow from a more general analysis of Vandermonde and Vandermonde-like matrices for which exact estimates for exponential ill-conditioning are developed in terms of logarithmic potentials.

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Rikhav Shah, John Urschel. 2026-04-16. On the exponential rate of the condition number of Fourier submatrices and Vandermonde matrices. https://arxiv.org/abs/2604.15135

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