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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. 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. 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. Major changes in this revised and expanded second edition: -Expansion of topics such as matrix functions, nonnegative matrices, and (unitarily invariant) matrix norms -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 new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.
Mathematicians and historians of mathematics and science will find in The Chinese Roots of Linear Algebra new ways to conceptualize the intellectual development of linear algebra.
Linear Algebra and Linear Models comprises a concise and rigorous introduction to linear algebra required for statistics followed by the basic aspects of the theory of linear estimation and hypothesis testing. The emphasis is on the approach using generalized inverses. Topics such as the multivariate normal distribution and distribution of quadratic forms are included. For this third edition, the material has been reorganised to develop the linear algebra in the first six chapters, to serve as a first course on linear algebra that is especially suitable for students of statistics or for those looking for a matrix theoretic approach to the subject. Other key features include: coverage of topics such as rank additivity, inequalities for eigenvalues and singular values; a new chapter on linear mixed models; over seventy additional problems on rank: the matrix rank is an important and rich topic with connections to many aspects of linear algebra such as generalized inverses, idempotent matrices and partitioned matrices. This text is aimed primarily at advanced undergraduate and first-year graduate students taking courses in linear algebra, linear models, multivariate analysis and design of experiments. A wealth of exercises, complete with hints and solutions, help to consolidate understanding. Researchers in mathematics and statistics will also find the book a useful source of results and problems.
Providing readers with the very basic knowledge necessary to begin research on differential equations with professional ability, the selection of topics here covers the methods and results that are applicable in a variety of different fields. The book is divided into four parts. The first covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part describes the basic results concerning linear differential equations, while the third deals with nonlinear equations. In the last part the authors write about the basic results concerning power series solutions. Each chapter begins with a brief discussion of its contents and history, and hints and comments for many problems are given throughout. With 114 illustrations and 206 exercises, the book is suitable for a one-year graduate course, as well as a reference book for research mathematicians.
Dealing with both the finite-dimensional and the compact theorems about triangularization, this volume will appeal to two different audiences. The first four chapters are of interest to linear-algebraists and other algebraists who use algebras or semigroups of matrices. These chapters include new approaches to some results that are widely used, as well as recent developments and extensions. The remaining chapters cover the structures of linear operators on Banach spaces will thus attract operator theorists. The authors obtain the most satisfactory results for algebras and semigroups of compact operators, although a number of important questions remain open; they hope that their presentation will stimulate further research. XXXXXXXX NEUER TEXT Written by well-respected authors in the field, this book deals with the finite-dimensional and compact theorems about triangularization. It is designed to appeal to mathematicians and students working in the fields of linear algebra or operator theory. Included are new approaches to some results that are widely used, as well as many recent developments and extensions. The presentation will surely stimulate further research into a number of important questions that remain open.
Introductory Probability is a pleasure to read and provides a fine answer to the question: How do you construct Brownian motion from scratch, given that you are a competent analyst? There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path.

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