Fuzhen Zhang - Böcker

What's in a name? To paraphrase Shakespeare's Juliet, that which - ilie Haynsworth called the Schur complement, by any other name would be just as beautiful. Nevertheless, her 1968 naming decision in honor of Issai Schur (1875-1941) has gained lasting acceptance by the mathematical com­ munity. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. Our goal is to expose the Schur complement as a rich and basic tool in mathematical research and applications and to discuss many significant re­ sults that illustrate its power and fertility. Although our book was originally conceived as a research reference, it will also be useful for graduate and up­ per division undergraduate courses in mathematics, applied mathematics, and statistics. The contributing authors have developed an exposition that makes the material accessible to readers with a sound foundation in linear algebra. The eight chapters of the book (Chapters 0-7) cover themes and varia­ tions 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 appli­ cations in statistics, probability, and numerical analysis. The chapters need not be read in the order presented, and the reader should feel at leisure to browse freely through topics of interest.

The aim of this text is to present fundamental ideas, results, and techniques concisely, mainly in matrix theory with some in linear algebra. The book contains ten chapters covering various topics ranging from rank, similarity, and special matrices, to Schur complements, matrix normality, and majorization. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. Major changes in this third edition include:expansion of topics such as eigenvalue continuity, matrix functions, nonnegative matrices, matrix norms, and majorizationinclusion of more than 200 examples and more than 1500 exercisesemphasis on basic techniques and skills for partitioned matrices through which a variety of matrix results and matrix inequalities are shownshowcase of many majorization-type inequalities for diagonal entries, eigenvalues, and singular values of matrices.This book can be used as a textbook or a supplement for a linear algebra and matrix theory class or a seminar for advanced undergraduate or graduate students. Prerequisites include a solid background in elementary linear algebra and calculus. The text can also serve as a reference for researchers in the field of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other scientific areas.From reviews of the second edition:The author has made a valuable contribution to the textbook literature on matrix theory, and his work will be appreciated by students and teachers of the subject. … is recommended reading for all those wishing to acquaint themselves with basic matrix theory.— Vicenţiu D. Rădulescu, Zentralblatt MATHIn many places several different proofs are presented for a theorem. This causes the book to be very attractive and readable. This book is useful for researchers as well as graduate students working in linear algebra, operator theory, statistics, computer science, engineering, applied mathematics, economics, and other disciplines.— Mohammad Sal Moslehian, Mathematical Reviews

What's in a name? To paraphrase Shakespeare's Juliet, that which - ilie Haynsworth called the Schur complement, by any other name would be just as beautiful. Nevertheless, her 1968 naming decision in honor of Issai Schur (1875-1941) has gained lasting acceptance by the mathematical com­ munity. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. Our goal is to expose the Schur complement as a rich and basic tool in mathematical research and applications and to discuss many significant re­ sults that illustrate its power and fertility. Although our book was originally conceived as a research reference, it will also be useful for graduate and up­ per division undergraduate courses in mathematics, applied mathematics, and statistics. The contributing authors have developed an exposition that makes the material accessible to readers with a sound foundation in linear algebra. The eight chapters of the book (Chapters 0-7) cover themes and varia­ tions 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 appli­ cations in statistics, probability, and numerical analysis. The chapters need not be read in the order presented, and the reader should feel at leisure to browse freely through topics of interest.

The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains ten chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. Major changes in this revised and expanded second edition: -Expansion of topics such as matrix functions, nonnegative matrices, and (unitarily invariant) matrix norms-The inclusion of more than 1000 exercises; -A new chapter, Chapter 4, with updated material on numerical ranges and radii, matrix norms, and special operations such as the Kronecker and Hadamard products and compound matrices-A new chapter, Chapter 10, on matrix inequalities, which presents a variety of inequalities on the eigenvalues and singular values of matrices and unitarily invariant norms.This book can be used as a textbook or a supplement for a linear algebra and matrix theory class or a seminar for senior undergraduate or graduate students. Prerequisites include a decent background in elementary linear algebra and calculus. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields.

This is the revised and expanded edition of the problem book Linear Algebra: Challenging Problems for Students, now entitled Problems in Linear Algebra and Matrix Theory. This new edition contains about fifty-five examples and many new problems, based on the author's lecture notes of Advanced Linear Algebra classes at Nova Southeastern University (NSU-Florida) and short lectures Matrix Gems at Shanghai University and Beijing Normal University.The book is intended for upper division undergraduate and beginning graduate students, and it can be used as text or supplement for a second course in linear algebra. Each chapter starts with Definitions, Facts, and Examples, followed by problems. Hints and solutions to all problems are also provided.