Generalized singular values and interlacing inequalities
type: Item
instance of: scholarly article
title: Generalized singular values and interlacing inequalities
zbMATH Open document ID: 0692.15015
DOI: 10.1016/0022-247X(89)90282-5
author: Harald K. Wimmer
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: If a complex matrix B has singular values \(\tau_ 1,\tau_ 2,...,\tau_ b\), let \(\| B\|_ g=(\sum^{b}_{i=1}| \tau_ i|^ p)^{1/p},\quad p\geq 1.\) The g-singular values of an \(m\times n\) complex matrix A, \(m\geq n\), are defined by \(\gamma_ k=\| A^{[k]}\|_ g/\| A^{[k-1]}\|_ g,\quad k=1,2,...,n,\) where \(A^{[k]}\) denotes the kth compound of A. The author proves the following interlocking inequalities: if \(\gamma_ k\), \(k=1,2,...,n\), are the g-singular values of A and if \(\lambda_ j\), \(j=1,2,...,n-1\), are the g-singular values of an \(m\times (n-1)\) matrix obtained by deleting any one column of A, then \(\gamma_ k\geq \lambda_ k\geq \gamma_{k+1},\quad k=1,2,...,n-1.\)
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 4132365
zbMATH Keywords: interlacing inequalities
zbMATH Keywords: complex matrix
zbMATH Keywords: singular values
zbMATH Keywords: interlocking inequalities
reviewed by: Peter Gibson
MaRDI profile type: Publication
cites work: Q5336417
cites work: Matrix Analysis
cites work: Q5343739
cites work: Interlacing inequalities for generalized singular values
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description: scientific article; zbMATH DE number 4132365