The spectral properties of Vandermonde matrices with clustered nodes

• Published in: Linear algebra and its applications Vol. 609; pp. 37 - 72
• Main Authors: Batenkov, Dmitry, Diederichs, Benedikt, Goldman, Gil, Yomdin, Yosef
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.01.2021; American Elsevier Company, Inc
• Subjects: Clusters; Condition number; Least squares method; Linear algebra; Matrices (mathematics); Nodes; Nonuniform Fourier matrices; Singular values; Spectrum analysis; Sub-Rayleigh resolution; Subspace angles; Subspaces; Super-resolution; Vandermonde matrices with nodes on the unit circle; Nonuniform Fourier matrices; Super-resolution; Singular values; Subspace angles; Condition number; Sub-Rayleigh resolution; Vandermonde matrices with nodes on the unit circle; primary
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2020.08.034

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 h≲1N, and the clusters being separated from each other by at least θ≳1N. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at mostπ2−c1Nθ−c2Nh, 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.