How to express a parallel sum of \(k\) matrices
type: Item
instance of: scholarly article
title: How to express a parallel sum of \(k\) matrices
zbMATH Open document ID: 0997.15004
author: Yongge Tian
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-24T00:00:00Z
review text: The paper deals with the parallel sum of \(k\) matrices. For two Hermitian nonnegative definite matrices \(A\) and \(B\) of the same size, the parallel sum is defined by means of the Moore-Penrose inverse as: \(A:B = A(A+B)^{\dagger}B.\) A natural extension to \(k\) matrices has not a simple form to be expressed. Then an alternative expression for \(A_1:A_2:\cdots:A_k \) is given in terms of block matrices. The proof of the main result is based on some rank equalities and Moore-Penrose inverse approach. Given two positive definite matrices \(A\) and \(B\), the equality \(A:B = (A^{-1}+B^{-1})^{-1}\) holds. In the paper, a neccesary and sufficient geometrical condition is derived for the equality \(A_1:A_2:\cdots:A_k = (A_1^{\dagger} + A_2^{\dagger} + \cdots + A_k^{\dagger})^{\dagger}\) for Hermitian nonnegative definite matrices of the same order \(n\).
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 1744279
zbMATH Keywords: parallel sum
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: rank
zbMATH Keywords: shorted matrix
reviewed by: Néstor Thome
MaRDI profile type: Publication
full work available at URL: jmaa.2001.7732
OpenAlex ID: W2091379119
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cites work: The Moore-Penrose inverses of \(m\times n\) block matrices and their applications
DOI: 10.1006/JMAA.2001.7732
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description: scientific article; zbMATH DE number 1744279