# The spectral properties of Vandermonde matrices with clustered nodes

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 ≲1 N, and the clusters being separated from each other by at least θ≳1 N. 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 π/2-c_1/N θ-c_2 N h, for some constants c_1,c_2 depending only on the multiplicities of the clusters. As a result, spectral analysis of V_N 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.

READ FULL TEXT