A generalization of Bergstrom's inequality and some applications
type: Item
instance of: scholarly article
title: A generalization of Bergstrom's inequality and some applications
zbMATH Open document ID: 0746.15011
DOI: 10.1016/0024-3795(92)90009-Y
author: Sze Fong Yau
author: Yoram Bresler
published in: Linear Algebra and its Applications
publication date: 1992-06-28T00:00:00Z
review text: The inequalities the authors considered are with respect to positive definite and semidefinite Hermitian matrices. They first derive a generalized Bergstrom's inequality and then use it to derive some other interesting results. The following notation is being used: \((\cdot)*=\) Hermitian transpose; \({\mathbf A}\odot{\mathbf B}=\) Schur-Hadamard (element by element) matrix product; \(A_{ij},({\mathbf A})_{ij}=i,j\) element of \({\mathbf A}\). Let \({\mathbf P},{\mathbf Q}\in{\mathcal C}^{N\times N}\) such that \({\mathbf P}>0\) and \({\mathbf Q}\geq 0\), \(Q_{ii}>0\). Then the authors prove the following tight inequalities: a) \(Q_{ii}[({\mathbf P}\odot{\mathbf Q})^{-1}]_{ii}\leq({\mathbf P}^{- 1})_{ii}\) ; b) \(Q_{ii}\{[\text{Re}({\mathbf P}\odot{\mathbf Q})]^{- 1}\}_{ii}\leq\max\{[\text{Re}({\Phi P\Phi}^*)]^{-1}\}_{ii}\) ; c) \(\text{det}({\mathbf P}\odot{\mathbf Q})\geq(\text{det}{\mathbf P})\prod^ N_{i=1}Q_{ii}\) ; d) \(\text{det}\text{Re}({\mathbf P}\odot{\mathbf Q})\geq\min\text{det}[\text{Re}({\Phi P\Phi}^*)]\prod^ N_{i=1}Q_{ii}\) , where in (b) and (d) maximization and minimization are over all unitary diagonal matrices \(\Phi\in{\mathcal C}^{N\times N}\). The paper contains the sections of generalization of Bergstrom's inequality, two Hadamard-product matrix inequalities, and generalization of Oppenheim's inequality.
reviewed by: Yueh-er Kuo
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 32817
zbMATH Keywords: determinant inequality
zbMATH Keywords: Schur-Hadamard matrix product
zbMATH Keywords: semidefinite Hermitian matrices
zbMATH Keywords: Bergstrom's inequality
zbMATH Keywords: Oppenheim's inequality
MaRDI profile type: Publication
cites work: Notes on Matrix Theory--IV (An Inequality Due to Bergstrom)
cites work: Q5816708
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description: scientific article; zbMATH DE number 32817