The Schur Complement and Its Applications - Numerical Methods and Algorithms 1st Ed. Softcover of Orig. Ed. 2005 edition

Jacket Description/Back: 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 (Universite 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. "Review Quotes: From the reviews of the first edition: "The book consists of eight chapters, each written by experts in their field, devoted to certain aspects and applications of the Schur complement. They can be read independently of each other. The book can serve as a research reference, as it contains many new results and results not yet appeared in books. The articles contain thorough expositions, so they can be understood by anyone having a good knowledge of linear algebra." (Ludwig Elsner, Zentralblatt MATH, Vol. 1075, 2006) Review Quotes: From the reviews of the first edition: "The book consists of eight chapters, each written by experts in their field, devoted to certain aspects and applications of the Schur complement. They can be read independently of each other. The book can serve as a research reference, as it contains many new results and results not yet appeared in books. The articles contain thorough expositions, so they can be understood by anyone having a good knowledge of linear algebra." (Ludwig Elsner, Zentralblatt MATH, Vol. 1075, 2006) Review Quotes: From the reviews of the first edition: "The book consists of eight chapters, each written by experts in their field, devoted to certain aspects and applications of the Schur complement. They can be read independently of each other. The book can serve as a research reference, as it contains many new results and results not yet appeared in books. The articles contain thorough expositions, so they can be understood by anyone having a good knowledge of linear algebra." (Ludwig Elsner, Zentralblatt MATH, Vol. 1075, 2006)"Review Quotes: From the reviews of the first edition: "The book consists of eight chapters, each written by experts in their field, devoted to certain aspects and applications of the Schur complement. They can be read independently of each other. The book can serve as a research reference, as it contains many new results and results not yet appeared in books. The articles contain thorough expositions, so they can be understood by anyone having a good knowledge of linear algebra." (Ludwig Elsner, Zentralblatt MATH, Vol. 1075, 2006)"Table of Contents: Preface -- Chapter 0. Historical Introduction: Issai Schur and the Early Development of the Schur Complement / Simo Puntanen, George P. H. Styan -- 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 / Roger A. Horn, Fuzhen Zhang -- 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 / Jianzhou Liu -- 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 / Fuzhen Zhang -- 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 / Charles R. Johnson, Ronald L. Smith -- 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 / Tsuyoshi Ando -- 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 / Simo Puntanen, George P. H. Styan -- 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 / Claude Brezinski -- 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.

Contributor Bio:  Zhang, Fuzhen Fuzhen Zhang is assistant professor in the Department of Mathematical Sciences at Nova Southeastern University in Fort Lauderdale, Florida.