Lower bounds for permanents of Gram matrices having a rank one principal submatrix
type: Item
instance of: scholarly article
title: Lower bounds for permanents of Gram matrices having a rank one principal submatrix
zbMATH Open document ID: 0658.15017
DOI: 10.1016/0024-3795(88)90063-8
author: Thomas H. Pate
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The author presents the following: Let \(A=[a_{ij}]\) be a positive semidefinite Hermitian \((n+p)\times (n+p)\) matrix partitioned into blocks \(A_{11}\), \(A_{12}\), \(A_{21}\), and \(A_{22}\) such that \(A_{11}\) is \(n\times n\) and \(A_{22}\) is \(p\times p\). Let \(\tilde A\) denote the \((n+p-1)\times (n+p-1)\) matrix get from A by deleting the first row and column of A. If \(A_{11}\) is of rank one, then \(per(A)\geq na_{11}per(\tilde A)\) with equality iff \(A_{12}\) is the zero matrix or A has a row of zeros. Some examples in this regard are added.
reviewed by: Seshadri Sridhar
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 4075289
zbMATH Keywords: lower bounds
zbMATH Keywords: permanents
zbMATH Keywords: Gram matrices
zbMATH Keywords: rank one principal submatrix
zbMATH Keywords: positive semidefinite Hermitian
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90063-8
OpenAlex ID: W1985036569
cites work: The Hadamard Theorem for Permanents
cites work: Q5520710
cites work: Norm of symmetric product compared with norm of tensor product
cites work: An inequality involving permanents of certain direct products
description: scientific article; zbMATH DE number 4075289