The Schur Complement and Its Applications

Simo Puntanen, George P. H. Styan (auth.), Fuzhen Zhang (eds.)9780387242712, 0-387-24271-6

The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. This book describes the Schur complement as a rich and basic tool in mathematical research and applications and discusses many significant results that illustrate its power and fertility. The eight chapters of the book cover themes and variations on the Schur complement, including its historical development, basic properties, eigenvalue and singular value inequalities, matrix inequalities in both finite and infinite dimensional settings, closure properties, and applications in statistics, probability, and numerical analysis. The chapters need not be read in order, and the reader should feel free to browse freely through topics of interest.

Although the book is primarily intended to serve as a research reference, it will also be useful for graduate and advanced undergraduate courses in mathematics, applied mathematics, and statistics. The contributing authors’ exposition makes most of the material accessible to readers with a sound foundation in linear algebra.

The book, edited by Fuzhen Zhang, was written by several distinguished mathematicians: T. Ando (Hokkaido University, Japan), C. Brezinski (Université des Sciences et Technologies de Lille, France), R. Horn (University of Utah, Salt Lake City, U.S.A.), C. Johnson (College of William and Mary, Williamsburg, U.S.A.), J.-Z. Liu (Xiangtang University, China), S. Puntanen (University of Tampere, Finland), R. Smith (University of Tennessee, Chattanooga, USA), and G.P.H. Steyn (McGill University, Canada). Fuzhen Zhang is a professor of Nova Southeastern University, Fort Lauderdale, U.S.A., and a guest professor of Shenyang Normal University, Shenyang, China.

Audience

This book is intended for researchers in linear algebra, matrix analysis, numerical analysis, and statistics.

Table of contents :
Historical Introduction: Issai Schur and the Early Development of the Schur Complement….Pages 1-16
Basic Properties of the Schur Complement….Pages 17-46
Eigenvalue and Singular Value Inequalities of Schur Complements….Pages 47-82
Block Matrix Techniques….Pages 83-110
Closure Properties….Pages 111-136
Schur Complements and Matrix Inequalities: Operator-Theoretic Approach….Pages 137-162
Schur complements in statistics and probability….Pages 163-226
Schur Complements and Applications in Numerical Analysis….Pages 227-258