Complementary Schur complements
type: Item
instance of: scholarly article
title: Complementary Schur complements
zbMATH Open document ID: 0795.15006
DOI: 10.1016/0024-3795(94)90507-X
author: Harm Bart
author: V. E. Tsekanovskii
published in: Linear Algebra and its Applications
publication date: 1994-04-19T00:00:00Z
review text: Let \(M\) and \(N\) be \(m\times m\) and \(n\times n\) complex matrices, respectively. The authors define \(M\) and \(N\) to be ``Schur coupled'' if there exist matrices \(A,B,C\) and \(D\) such that \(M=D-CA^{-1}B\) and \(N=A- BD^{-1}C\). The main result of this paper is that the following are equivalent: (i) \(M\) and \(N\) are Schur coupled; (ii) there is an \((m+n)\times(m+n)\) matrix \(H\) such that the leading \(m\times m\) principal submatrix of \(H\) is \(M\) and the final \(n\times n\) principal submatrix of \(H^{-1}\) is \(N\); and (iii) \(m-\text{rank} M=n-\text{rank} N\).
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 553979
zbMATH Keywords: complementary Schur complements
zbMATH Keywords: Schur coupled matrices
reviewed by: John D. Dixon
MaRDI profile type: Publication
cites work: Addendum to: The coupling method for solving integral equations
cites work: Q4287397
cites work: What are Schur complements, anyway?
cites work: Completing a matrix when certain entries of its inverse are specified
cites work: Equivalence, linearization, and decomposition of holomorphic operator functions
cites work: A note on matrix inversion
description: scientific article; zbMATH DE number 553979