Convexity of the inverse and Moore-Penrose inverse
type: Item
instance of: scholarly article
title: Convexity of the inverse and Moore-Penrose inverse
zbMATH Open document ID: 1215.15004
author: Kenneth Nordström
published in: Linear Algebra and its Applications
publication date: 2011-03-04T00:00:00Z
review text: Based on a re-examination of the convexity results for the inverse of positive definite matrices, and the Moore-Penrose inverse of nonnegative definite matrices, a more general concept of strong convexity is defined which provides additional information on the convexity behaviour and geometry of matrix functions known to be strictly convex. Convexity results which include characterisations of the null spaces are proved and further used in order to obtain short proofs on sums of reproducing kernels, to derive a number of matrix-mean-type inequalities, and, subsequently, a quasilinear extremal representation of a weighted harmonic matrix mean is derived from these inequalities. The main part of the paper deals with a detailed study of convexity of Moore-Penrose inverse of nonnegative definite matrices providing extensions and generalisations of all the earlier work in this area, many of the obtained results being straightforward extensions to the case of bounded linear operators acting on an infinite-dimensional Hilbert space.
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 5860479
zbMATH Keywords: extremal representation
zbMATH Keywords: indefinite inner product
zbMATH Keywords: matrix mean
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: strongly convex matrix function
zbMATH Keywords: positive definite matrices
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: matrix-mean-type inequalities
MaRDI profile type: Publication
full work available at URL: j.laa.2010.11.023
OpenAlex ID: W2062432742
cites work: Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses
cites work: Series and parallel addition of matrices
cites work: Shorted Operators. II
cites work: Q4169818
cites work: Concavity of certain maps on positive definite matrices and applications to Hadamard products
cites work: Majorizations and inequalities in matrix theory
cites work: Theory of Reproducing Kernels
cites work: Q5333457
cites work: Inequalities between means of positive operators
cites work: Q4774569
cites work: A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse
cites work: The rank of a difference of matrices and associated generalized inverses
cites work: Q4405419
cites work: Q3245669
cites work: A potential-theoretic approach to parallel addition
cites work: Q3958598
cites work: G-majorization with applications to matrix orderings
cites work: Q5812325
cites work: Q4284368
cites work: Applications of an Inequality for the Schur Complement
cites work: Q4040931
cites work: On the matrix convexity of the moore—penrose inverse and some applications
cites work: Q3844024
cites work: Means of positive linear operators
cites work: Some results on matrix monotone functions
cites work: Some operator inequalities of the Schwarz type
cites work: An Inequality for Linear Transformations
cites work: Inequalities: theory of majorization and its applications
cites work: On the simultaneous diagonalization of two semi-definite matrices
cites work: Some further aspects of the löwner-ordering antitonicity of the moore-penrose inverse
cites work: A Multivariate Tchebycheff Inequality
cites work: Q4849481
cites work: Q3326646
cites work: Positive definite Hermitian matrices and reproducing kernels
cites work: Q4040547
cites work: A question regarding inequalities between Hermitian operators
cites work: Q3922026
cites work: A MULTIVARIATE GENERALIZATION OF TCHEBICHEV'S INEQUALITY
DOI: 10.1016/J.LAA.2010.11.023
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description: scientific article; zbMATH DE number 5860479