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2000•SIAM Journal on Matrix Analysis and ApplicationsRequires access

Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk

Fermín S. V. Bazán

Open publisher page 63 citations

Abstract

Let W N =W N (z 1 ,z 2 , . . . z 1 ) be a rectangular Vandermonde matrix of order n × N, $N\geq n,$ with distinct nodes z j in the unit disk and $z_j^{k-1}$ as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of W N is analyzed and bounds for the spectral condition number $\kappa_2(W_N)$ are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of W N is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.

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What this paper is about

Let W N =W N (z 1 ,z 2 , . . . z 1 ) be a rectangular Vandermonde matrix of order n × N, $N\geq n,$ with distinct nodes z j in the unit disk and $z_j^{k-1}$ as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of W N is analyzed and bounds for the spectral condition number $\kappa_2(W_N)$ are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of W N is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.

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Available abstract

Let W N =W N (z 1 ,z 2 , . . . z 1 ) be a rectangular Vandermonde matrix of order n × N, $N\geq n,$ with distinct nodes z j in the unit disk and $z_j^{k-1}$ as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of W N is analyzed and bounds for the spectral condition number $\kappa_2(W_N)$ are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of W N is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.

Key concepts: Vandermonde matrix, Mathematics, Unit disk, Combinatorics, Unit circle, Matrix (chemical analysis), Condition number, Unit (ring theory)

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