The Schur complement and its applications /

The Schur complement and its applications /

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Bibliographic Details
Imprint:New York : Springer Science, c2005.
Description:xvi, 295 p. ; 25 cm.
Language:English
Series:Numerical methods and algorithms ; v. 4
Subject:
Schur complement -- Data processing.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5643067
Hidden Bibliographic Details
Other authors / contributors:Zhang, Fuzhen.
ISBN:0387242716 (alk. paper)
Notes:Includes bibliographical references (p. [259]-288) and index.
Table of Contents:
  • Preface
  • Chapter 0. Historical Introduction: Issai Schur and the Early Development of the Schur Complement
  • 0.0. Introduction and mise-en-scene
  • 0.1. The Schur complement: the name and the notation
  • 0.2. Some implicit manifestations in the 1800s
  • 0.3. The lemma and the Schur determinant formula
  • 0.4. Issai Schur (1875-1941)
  • 0.5. Schur's contributions in mathematics
  • 0.6. Publication under J. Schur
  • 0.7. Boltz 1923, Lohan 1933, Aitken 1937 and the Banchiewicz inversion formula 1937
  • 0.8. Frazer, Duncan & Collar 1938, Aitken 1939, and Duncan 1944
  • 0.9. The Aitken block-diagonalization formula 1939 and the Guttman rank additivity formula 1946
  • 0.10. Emilie Virginia Haynsworth (1916-1985) and the Haynsworth inertia additivity formula
  • Chapter 1. Basic Properties of the Schur Complement
  • 1.0. Notation
  • 1.1. Gaussian elimination and the Schur complement
  • 1.2. The quotient formula
  • 1.3. Inertia of Hermitian matrices
  • 1.4. Positive semidefinite matrices
  • 1.5. Hadamard products and the Schur complement
  • 1.6. The generalized Schur complement
  • Chapter 2. Eigenvalue and Singular Value Inequalities of Schur Complements
  • 2.0. Introduction
  • 2.1. The interlacing properties
  • 2.2. Extremal characterizations
  • 2.3. Eigenvalues of the Schur complement of a product
  • 2.4. Eigenvalues of the Schur complement of a sum
  • 2.5. The Hermitian case
  • 2.6. Singular values of the Schur complement of a product
  • Chapter 3. Block Matrix Techniques
  • 3.0. Introduction
  • 3.1. Embedding approach
  • 3.2. A matrix inequality and its applications
  • 3.3. A technique by means of 2 x 2 block matrices
  • 3.4. Liebian functions
  • 3.5. Positive linear maps
  • Chapter 4. Closure Properties
  • 4.0. Introduction
  • 4.1. Basic theory
  • 4.2. Particular classes
  • 4.3. Singular principal minors
  • 4.4. Authors' historical notes
  • Chapter 5. Schur Complements and Matrix Inequalities: Operator-Theoretic Approach
  • 5.0. Introduction
  • 5.1. Schur complement and orthoprojection
  • 5.2. Properties of the map A [map to] [M]A
  • 5.3. Schur complement and parallel sum
  • 5.4. Application to the infimum problem
  • Chapter 6. Schur Complements in Statistics and Probability
  • 6.0. Basic results on Schur complements
  • 6.1. Some matrix inequalities in statistics and probability
  • 6.2. Correlation
  • 6.3. The general linear model and multiple linear regression
  • 6.4. Experimental design and analysis of variance
  • 6.5. Broyden's matrix problem and mark-scaling algorithm
  • Chapter 7. Schur Complements and Applications in Numerical Analysis
  • 7.0. Introduction
  • 7.1. Formal orthogonality
  • 7.2. Pade application
  • 7.3. Continued fractions
  • 7.4. Extrapolation algorithms
  • 7.5. The bordering method
  • 7.6. Projections
  • 7.7. Preconditioners
  • 7.8. Domain decomposition methods
  • 7.9. Triangular recursion schemes
  • 7.10. Linear control
  • Bibliography
  • Notation
  • Index
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