Some inequalities on generalized Schur complements

Bo Ying Wang; Xiuping Zhang; Fuzhen Zhang

doi:10.1016/S0024-3795(99)00071-3

Some inequalities on generalized Schur complements

Bo Ying Wang
, Xiuping Zhang
, Fuzhen Zhang

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

This paper presents some inequalities on generalized Schur complements. Let A be an n × n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A⁺ be the Moore-Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A⁺(α′)) ≥ λ((A/α)⁺), where α′ is the complement of α in {1,2,...,n}. 2. λ(A^r/α) ≤ λ^r(A/α.) for any real number r ≥ 1. 3. (C*AC)/α≤C*/α A(α′) C/α for any matrix C of certain properties on partitioning.

Original language	English
Pages (from-to)	163-172
Number of pages	10
Journal	Linear Algebra and its Applications
Volume	302-303
DOIs	https://doi.org/10.1016/S0024-3795(99)00071-3
State	Published - Dec 1 1999

ASJC Scopus Subject Areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Keywords

Generalized inverse
Positive semidefinite matrix
Schur complement

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10.1016/S0024-3795(99)00071-3

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title = "Some inequalities on generalized Schur complements",

abstract = "This paper presents some inequalities on generalized Schur complements. Let A be an n × n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore-Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A+(α′)) ≥ λ((A/α)+), where α′ is the complement of α in \{1,2,...,n\}. 2. λ(Ar/α) ≤ λr(A/α.) for any real number r ≥ 1. 3. (C*AC)/α≤C*/α A(α′) C/α for any matrix C of certain properties on partitioning.",

keywords = "Generalized inverse, Positive semidefinite matrix, Schur complement",

author = "Wang, \{Bo Ying\} and Xiuping Zhang and Fuzhen Zhang",

year = "1999",

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doi = "10.1016/S0024-3795(99)00071-3",

language = "English",

volume = "302-303",

pages = "163--172",

journal = "Linear Algebra and its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

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TY - JOUR

T1 - Some inequalities on generalized Schur complements

AU - Wang, Bo Ying

AU - Zhang, Xiuping

AU - Zhang, Fuzhen

PY - 1999/12/1

Y1 - 1999/12/1

N2 - This paper presents some inequalities on generalized Schur complements. Let A be an n × n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore-Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A+(α′)) ≥ λ((A/α)+), where α′ is the complement of α in {1,2,...,n}. 2. λ(Ar/α) ≤ λr(A/α.) for any real number r ≥ 1. 3. (C*AC)/α≤C*/α A(α′) C/α for any matrix C of certain properties on partitioning.

AB - This paper presents some inequalities on generalized Schur complements. Let A be an n × n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A+ be the Moore-Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A+(α′)) ≥ λ((A/α)+), where α′ is the complement of α in {1,2,...,n}. 2. λ(Ar/α) ≤ λr(A/α.) for any real number r ≥ 1. 3. (C*AC)/α≤C*/α A(α′) C/α for any matrix C of certain properties on partitioning.

KW - Generalized inverse

KW - Positive semidefinite matrix

KW - Schur complement

UR - https://www.scopus.com/pages/publications/0033269615

UR - https://www.scopus.com/pages/publications/0033269615#tab=citedBy

U2 - 10.1016/S0024-3795(99)00071-3

DO - 10.1016/S0024-3795(99)00071-3

M3 - Article

AN - SCOPUS:0033269615

SN - 0024-3795

VL - 302-303

SP - 163

EP - 172

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

ER -

Some inequalities on generalized Schur complements

Abstract

ASJC Scopus Subject Areas

Keywords

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