Diagonally scaled permutations and circulant matrices
type: Item
instance of: scholarly article
title: Diagonally scaled permutations and circulant matrices
zbMATH Open document ID: 0815.15008
DOI: 10.1016/0024-3795(94)90413-8
author: Jeffrey Stuart
author: James R. Weaver
published in: Linear Algebra and its Applications
publication date: 1995-03-09T00:00:00Z
review text: The following results are obtained. (1) If an \(n \times n\) complex matrix \(R = D_ 1P\) is given, where \(D_ 1\) is a complex diagonal matrix, and \(P\) is a permutation matrix, then \(R\) is permutation similar to the direct sum of matrices of the form \(DC\), where \(D\) is a complex diagonal matrix and \(C\) is a circulant permutation matrix. (2) The authors consider two cases for \(R = DC\): when the scaling matrix \(D\) is nonsingular, and when \(D\) is singular. For the singular case, the authors prove that \(R\) is nilpotent, and obtain upper and lower bounds on the index of nilpotency of \(R\). In addition, the authors represent an arbitrary \(n\times n\) Toeplitz matrix as a sum of matrices of the form \(D(k, \alpha, \beta) C^ k\) for \(k = 1, \dots, n\), where \(D(k, \alpha, \beta)\) is a diagonal matrix.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 720486
zbMATH Keywords: factorization
zbMATH Keywords: circulant matrix
zbMATH Keywords: complex matrix
zbMATH Keywords: diagonal matrix
zbMATH Keywords: permutation matrix
zbMATH Keywords: scaling matrix
zbMATH Keywords: index of nilpotency
zbMATH Keywords: Toeplitz matrix
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90413-8
OpenAlex ID: W2008220454
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description: scientific article; zbMATH DE number 720486