Factorization of finite rank Hankel and Toeplitz matrices
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.
scientific article; zbMATH DE number 64135
scientific article