Generalizations of two inequalities involving Hermitian forms
type: Item
instance of: scholarly article
title: Generalizations of two inequalities involving Hermitian forms
zbMATH Open document ID: 0574.15009
DOI: 10.1016/0024-3795(85)90096-5
author: Sin-Chung Chang
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Let \(\lambda_ 1\) and \(\lambda_ N\) be, respectively, the greatest and smallest eigenvalues of the Hermitian matrix \(H=(h_{ij})\). For x with \((x,x)=1\) and H positive definite the Kantorovich inequality states \[ (1)\quad (\lambda_ 1+\lambda_ N)^ 2/4\lambda_ 1\cdot \lambda_ N\geq (x,Hx)(x,H^{-1}x)\geq 1. \] For \(| y_ i| \leq 1\), \(i=1,...,N\) consider the Hermitian matrix \(M(y):=(\delta_{ij}+(1- \delta_{ij})\bar y_ iy_ j)\) and the Hadamard products M(y)*H and \(M(y)*H^{-1}\). It is shown that the inequality (1) remains true for M(y)*H and \(M(y)*H^{-1}\) instead of H and \(H^{-1}\); furthermore the estimate \(\rho\) (M(y)*H)\(\leq \rho (H)\) for the spectral radius \(\rho\) (M(y)*H) of M(y)*H is given.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 3916478
zbMATH Keywords: Hermitian form
zbMATH Keywords: eigenvalue bounds
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: Kantorovich inequality
zbMATH Keywords: Hadamard products
zbMATH Keywords: estimate
zbMATH Keywords: spectral radius
reviewed by: Hans-Jochen Bartels
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90096-5
OpenAlex ID: W2005225567
cites work: Q5820676
cites work: On a Generalization of an Inequality of L. V. Kantorovich
cites work: Q5331504
Recommended article: Q4317080
Recommended article: Q5438000
Recommended article: Q2935669
Recommended article: Generalizations of some Hermite-Hadamard-type inequalities
Recommended article: A Buzano type inequality for two Hermitian forms and applications
Recommended article: Generalized Hermite-Hadamard type integral inequalities
Recommended article: Q5003008
Recommended article: Q5361075
Recommended article: Q6071582
description: scientific article; zbMATH DE number 3916478