The spectral properties of Vandermonde matrices with clustered nodes

Dmitry Batenkov, Benedikt Diederichs, Gil Goldman, Yosef Yomdin

Research output: Contribution to journalArticlepeer-review

Abstract

We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most [Formula presented], and the clusters being separated from each other by at least [Formula presented]. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at most [Formula presented] for some constants c1,c2 depending only on the multiplicities of the clusters. As a result, spectral analysis of VN is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of V, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with s.

Original languageEnglish
Pages (from-to)37-72
Number of pages36
JournalLinear Algebra and Its Applications
Volume609
DOIs
StatePublished - 15 Jan 2021

Keywords

  • Condition number
  • Nonuniform Fourier matrices
  • Singular values
  • Sub-Rayleigh resolution
  • Subspace angles
  • Super-resolution
  • Vandermonde matrices with nodes on the unit circle

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