Matrix Theory

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.

• Hermitian matrices
• compound matrices
• contractions
• linear algebra
• majorization inequalities
• matrix decompositions
• matrix functions
• matrix inequalities
• matrix polynomials
• matrix theory
• normal matrices
• partitioned matrices
• positive semidefinite matrices
• unitary matrices

From the reviews of the second edition:

“This book provides an interesting and recommendable text for a second course on linear algebra with emphasis on the theory of matrices.” (H. Mitsch, Monatshefte für Mathematik, Vol. 169 (1), January, 2013)

“For the practitioner, someone new to the field, or users of matrices from other areas, there is a terse, but friendly, discussion of a number of useful topics, a few of which are not otherwise in book form. … It is nice and friendly to read, with a simple, direct style.” (Charles Johnson, SIAM Review, Vol. 55 (1), 2013)

“The main purpose of this volume is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. … Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. 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 MATH, Vol. 1229, 2012)

“This interesting book is the revised version … covering materials from the basic elements of matrix theory to more advanced topics. … Each chapter starts with an introduction and includes a lot of well-selected problems … . In 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, Issue 2012 h)

• Dept. of Mathematics, Nova Southeastern University, Fort Lauderdale, USA

  Fuzhen Zhang

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