New lower bounds on eigenvalue of the Hadamard product of an \(M\)-matrix and its inverse
type: Item
instance of: scholarly article
title: New lower bounds on eigenvalue of the Hadamard product of an \(M\)-matrix and its inverse
zbMATH Open document ID: 1163.15019
author: Yaotang Li
author: Fubin Chen
author: De-Feng Wang
published in: Linear Algebra and its Applications
publication date: 2009-02-10T00:00:00Z
review text: The main results which improve some earlier ones are the following. Let \(A= (a_{ij}) \in\mathbb R^{n \times n} \) be an \(M\)-matrix and \(\tau (A \circ A^{-1})\) be the minimum eigenvalue of the Hadamard product of \(A\) and \(A^{-1}\). Then \(\tau (A \circ A^{-1}) \geqslant \min _i\{{1-\frac{1} {{a_{ii}}}\sum_{j \neq i} {|{a_{ji} }|m_{ji}}}\}\), where \(m_{ji}= \frac{{|{a_{ji}}|+ \sum_{k \neq j,i} {|{a_{jk}}|r_i }}} {{a_{jj}}}\), \(j \neq i\), \(j\in\mathbb N\); \(r_i= \max _{l \neq i} \frac{{|{a_{li}}|}} {{|{a_{ll}}|- \sum_{k\neq l,i} {|{a_{lk}}|}}}\). In addition let \(A^{-1}\) be doubly stochastic. Then \(\tau(A\circ A^{-1}) \geqslant \min _i \{{\frac{{a_{ii}-m_i \sum_{k\neq i} {|{a_{ik}}|}}} {{1+ \sum_{j \neq i} {m_{ji}}}}}\}\).
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5505638
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: Hadamard product
zbMATH Keywords: minimum eigenvalue
zbMATH Keywords: lower bounds
zbMATH Keywords: doubly stochastic
MaRDI profile type: Publication
full work available at URL: j.laa.2008.11.002
OpenAlex ID: W1973342845
cites work: Q5604166
cites work: An inequality for the Hadamard product of an M-matrix and an inverse M- matrix
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cites work: Proof of a conjecture of Fiedler and Markham
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cites work: Lower bounds for the minimum eigenvalue of Hadamard product of an \(M\)-matrix and its inverse
cites work: Some inequalities for the Hadamard product and the Fan product of matrices
cites work: Minimal Gerschgorin sets
cites work: On a conjecture of Fiedler and Markham
cites work: A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices
DOI: 10.1016/J.LAA.2008.11.002
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description: scientific article; zbMATH DE number 5505638