×

Matrix analysis and applications. (English) Zbl 1381.65027

Cambridge: Cambridge University Press (ISBN 978-1-108-41741-9/hbk; 978-1-108-27758-7/ebook). xxvi, 723 p. (2017).
This is a book offering a broad selection of topics on the theory, methods and applications of matrix analysis. Alongside the core subjects in matrix analysis, such as matrix algebra, the solution of linear systems, and matrix eigenvalue problems, the author introduces new recent applications and perspectives that are unique to this book, such as sparse representation, gradient analysis, matrix optimization, low-rank plus sparse decomposition of a matrix, matrix completion, and nonnegative factorization of a matrix. Also included are subspace analysis and tensor analysis, subjects which are often neglected in other books.
This book consists of ten chapters, spread over three parts, covers the core theory and methods in matrix analysis, and places particular emphasis on the recent applications in various science and engineering disciplines. Each chapter is concerned with one topic, begins with classical results and treats in some depth the recent results that have been proved to be important to applied mathematics as well as fertile fields. The exercises included in each chapter are plentiful, well thought-out and definitely aid the reader in understanding the subject and its applications.
Part I is on matrix algebra: it contains Chapters 1 through 3 and focuses on the elementary material. Chapter 1 is an introduction to matrix algebra. Chapter 2 presents some special matrices, such as the Toeplitz matrix and Hadamard matrix, that are often met in real-world applications. Chapter 3 is devoted to matrix differential, which is an important tool in matrix optimization.
Part II is on matrix analysis: this is the heart of this book and consists of six chapters. Chapter 4 is devoted to the gradient analysis of functions in vector/matrix, with applications in matrix optimization. Chapter 5 presents a comprehensive study of singular values of matrices, including the singular value decomposition (SVD) of a matrix, the generalized SVD of a matrix-pencil, low-rank plus sparse decomposition of a matrix and matrix completion. Chapter 6 focuses on methods for linear matrix equations \(AX=B\), includes the Tikhonov regularization, the total least squares, nonnegative matrix factorization, and methods for sparse solution. Chapter 7 deals with eigenvalue problems. This chapter starts from matrix eigenvalue decomposition (EVD) and then presents various generalizations of EVD: the generalized eigenvalue problem, the generalized Rayleigh quotient, the quadratic eigenvalue problem, and the joint diagonalization of several matrices. Chapter 8 focuses on theoretical analysis of subspaces and some applications of subspace methods. Chapter 9 is devoted to orthogonal and oblique projectors and their applications.
Part III is on tensor analysis and consists simply of Chapter 10. The contents of this chapter include: the basic algebraic operations, representation as matrices, Tucker decomposition, parallel factor decomposition, nonnegative tensor decomposition and tensor completion, together with interesting applications.
The structure of this book is congruent with the author’s goal of writing a book that “presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light”. The background required of the reader is a good knowledge advanced calculus. The proof of most results are omitted mainly because of the limitation of space. This book presents a modern treatment of a broad range of subjects in matrix analysis. It may be used as a self-contained reference for a variety of audiences.

MSC:

65Fxx Numerical linear algebra
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65K05 Numerical mathematical programming methods
PDFBibTeX XMLCite
Full Text: DOI