Bounds for the infinity norm of the inverse for certain \(M\)- and \(H\)-matrices
type: Item
instance of: scholarly article
title: Bounds for the infinity norm of the inverse for certain \(M\)- and \(H\)-matrices
zbMATH Open document ID: 1158.15019
author: L. Yu Kolotilina
published in: Linear Algebra and its Applications
publication date: 2008-12-12T00:00:00Z
review text: Let \(\bigcup_{i=1}^n M_i\) be a partition of the index set \(\{1,\dots, n\}\) into disjoint nonempty subsets. For an \(m\times m\) complex matrix \(A\), let \(r_i(A)\) be the \(i\)th row sum of the entries of \(A\), denote \(A_{ij}=A(M_i,M_j)\), for \(i,j=1,\dots,n\), the matrix that lies in the rows of \(A\) indexed by \(M_i\) and the columns indexed by \(M_j\), and consider the \(n\times n\) matrix \(A^{(k_1,k_2,\dots,k_n)}=(r_{k_i}(A_{ij}))\). The author shows that if \(A\) is a partitioned \(M\)-matrix, then it is nonsingular, and \(A^{-1}\) verifies the inequalities \[ \min_{k_1,\dots,k_n}\| (A^{(k_1,\dots,k_n)})^{-1}\| _\infty \leqslant \| A^{-1}\| _\infty \leqslant \max_{k_1,\dots,k_n}\| (A^{(k_1,\dots,k_n)})^{-1}\| _\infty . \] A condition for the equality case is presented. An analogous result is established for an upper bound for partitioned \(H\)-matrices and the main result is compared with others already known.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5382410
zbMATH Keywords: infinity norm
zbMATH Keywords: inverse matrix
zbMATH Keywords: PM-matrices
zbMATH Keywords: PH-matrices
zbMATH Keywords: M-matrices
zbMATH Keywords: H-matrices
zbMATH Keywords: strictly diagonally dominant matrices
reviewed by: C. M.da Fonseca
MaRDI profile type: Publication
full work available at URL: j.laa.2008.09.005
OpenAlex ID: W2106537516
cites work: Convergence Properties of the Spline Fit
cites work: Monotonicity and Discretization Error Estimates
cites work: Q5725702
cites work: Q4040931
cites work: Estimation of \(\| A ^{-1} \|_\infty \) and the smallest singular value
cites work: The infinity norm bound for the inverse of nonsingular diagonal dominant matrices
cites work: Upper bounds for the infinity norm of the inverse of SDD and \(\mathcal S\)-SDD matrices
cites work: Bounds for norms of the matrix inverse and the smallest singular value
cites work: On Two-Sided Bounds Related to Weakly Diagonally Dominant M-Matrices with Application to Digital Circuit Dynamics
cites work: A lower bound for the smallest singular value of a matrix
cites work: On diagonal dominance arguments for bounding \(\| A^{-1}\|_\infty\)
DOI: 10.1016/J.LAA.2008.09.005
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description: scientific article; zbMATH DE number 5382410