Notes on matrices with diagonally dominant properties
type: Item
instance of: scholarly article
title: Notes on matrices with diagonally dominant properties
zbMATH Open document ID: 1225.15007
author: F. O. Farid
published in: Linear Algebra and its Applications
publication date: 2011-08-26T00:00:00Z
review text: Let \(A \in \mathbb{C}^{n \times n}\) and \(a_{ij}\) denote its entries. For \(1 \leq k \leq n\), define \(r_k(A)\) by \(r_k(A) = \sum_{j=1,~j \neq k}^n~|a_{ij}|\) and the set \(J(A)\) by \(J(A)=\{i: 1 \leq i \leq n\) and \(|a_{ii}| > r_i(A)\}\). We say that \(A\) is diagonally dominant if \(|a_{jj}| \geq r_j(A)\) for all \(j\). \(A\) is said to be stricly diagonally dominant if \(J(A)=\{1,2,\dots,n\}\). \(A\) is called strictly generalized diagonally dominant if there exists a diagonal matrix \(Y\) such that \(AY\) is strictly diagonally dominant. For a nonempty proper subset \(S\) of \(\{1,2,\dots,n\}\), define \(r_k^S(A)\) by \(r_k^S(A) = \sum_{j \in S,~j \neq k}^n~|a_{kj}|\). A partition \(S_1,S_2\) of \(\{1,2,\dots,n\}\) is called a separation of \(\{1,2,\dots,n\}\), denoted by \((S_1,S_2)\). For a separation \((S_1,S_2)\), define \(f_A^{S_1}:S_1 \times S_2 \longrightarrow \mathbb{R}\) by \(f_A^{S_1}(i,j) = (|a_{ii}| - r_i^{S_1}(A)) (|a_{jj}| - r_j^{S_2}(A)) - r_i^{S_2}(A) r_j^{S_1}(A)\). \(A\) is called generalized doubly diagonally dominant if \(J(A)\) is nonempty and there exists a separation \((S_1,S_2)\) such that \(f_A^{S_1} \geq 0\). \(A\) is called strictly generalized doubly diagonally dominant if strict inequality holds in the inequality as above. As usual, the directed graph of \(A \in \mathbb{C}^{n \times n}\) is the directed graph on \(n\) distinct points called its vertices, denoted by \(v_1,\dots,v_n\), such that there is a directed arc from \(v_i\) to \(v_j\) if and only if \(a_{ij} \neq 0\). \(A\) is called cyclically diagonally dominant if for every cycle \(\varGamma\) in the graph of \(A\), the following inequality holds: \(\prod_{v_i \in \varGamma} |a_{ii}| \geq \prod_{v_i \in \varGamma} r_i(A)\). The author analyzes the relationship between certain classes of matrices with variants of the diagonal dominance property. The following are the main results of the paper: A sufficient condition for a generalized doubly diagonally dominant matrix t…
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 5943572
zbMATH Keywords: strictly generalized diagonally dominant matrix
zbMATH Keywords: generalized doubly diagonally dominant matrix
zbMATH Keywords: cyclically diagonally dominant matrix
zbMATH Keywords: directed graph
MaRDI profile type: Publication
full work available at URL: j.laa.2011.04.045
OpenAlex ID: W1976521968
cites work: Limits for the characteristic roots of a matrix. II
cites work: Matrices eigenvalues, and directed graphs
cites work: New criteria for identifying \(H\)-matrices
cites work: Criteria for invertibility of diagonally dominant matrices
cites work: Topics on a generalization of Gershgorin's theorem
cites work: Simple criteria for nonsingular H-matrices.
cites work: Criteria for generalized diagonally dominant matrices and \(M\)-matrices
cites work: Criteria for generalized diagonally dominant matrices and \(M\)-matrices. II
cites work: Q4479208
cites work: Matrix Analysis
cites work: Regions containing eigenvalues of a matrix
cites work: Doubly diagonally dominant matrices
cites work: The Schur complements of generalized doubly diagonally dominant matrices
cites work: A Sufficient Condition for Nonvanishing of Determinants
cites work: Q4829998
cites work: A note on A. Brauer's theorem
DOI: 10.1016/J.LAA.2011.04.045
description: scientific article; zbMATH DE number 5943572