Factorization of finite rank Hankel and Toeplitz matrices
type: Item
instance of: scholarly article
title: Factorization of finite rank Hankel and Toeplitz matrices
zbMATH Open document ID: 0753.15008
DOI: 10.1016/0024-3795(92)90420-F
author: Robert L. Ellis
author: David C. Lay
published in: Linear Algebra and its Applications
publication date: 1992-09-27T00:00:00Z
review text: The authors generalize some classical results concerning factorization of Hankel and Toeplitz matrices. Specifically, a classical result concerning Hankel matrices is: an \(n\times n\) positive semidefinite real Hankel matrix \(H\) with \(\text{rank} m\geq n-1\) can be factored as \(H=RDR^ T\), where \(D\) is an \(m\times n\) diagonal matrix with positive diagonal entries and \(R\) is an \(n\times m\) real Vandermonde matrix \([\xi_ k^{j-1}]_{1\leq j\leq n,1\leq k\leq m}\). The authors generalize the result by removing the conditions that \(H\) be positive semidefinite, real and of rank at least \(n-1\). In this generalization, the matrix \(R\) is replaced by a ``generalized'' Vandermonde matrix (where multiplicities are allowed in the generating polynomial) and the matrix \(D\) is replaced by a block-diagonal matrix whose diagonal blocks are reversed upper triangular, i.e., have zero entries in the lower right part of the matrix. A similar classical result concerning Toeplitz matrices is: an \(n\times n\) positive semidefinite Toeplitz matrix \(T\) with rank \(m\) has a factorization of the form \(T=RDR^*\) where \(D\) is an \(m\times m\) diagonal matrix with positive diagonal entries and \(R\) is an \(n\times m\) Vandermonde matrix \([\alpha_ k^{j-1}]_{1\leq j\leq n, 1\leq k\leq m}\) with \(|\alpha_ k|=1\) for \(1\leq k\leq m\). The authors' generalization of the Toeplitz theorem is similar to that for the Hankel case; the restriction that \(T\) be positive semidefinite is removed and the ``generalized'' Vandermonde matrix \(R\) and block diagonal matrix \(D\) are as in the generalization for the Hankel case. Preliminary results involve some factorization theorems for infinite Hankel and Toeplitz matrices of finite rank. Then the authors prove the existence of rank-preserving extensions of finite rank Hankel and Toeplitz matrices to infinite Hankel or Toeplitz matrices having the desired factorization property.
reviewed by: Joseph A. Ball
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 64135
zbMATH Keywords: factorization
zbMATH Keywords: Hankel matrix
zbMATH Keywords: Vandermonde matrix
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: positive semidefinite
zbMATH Keywords: rank-preserving extensions
zbMATH Keywords: finite rank
zbMATH Keywords: infinite Hankel or Toeplitz matrices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 64135