Browse Books

Reviewer: Jesse Louis Barlow

There are few computational algorithms in this book; the authors are interested in what matrices represent. For computational scientists or scientists for whom computation is an important part of their work, a matrix is something that is used to perform a computation. However, computational matrix analysts should be aware of R. W. Hamming's saying that the goal of computing is not numbers, but insight. To have informed insight, it is important to understand what is being computed. From that perspective, this book is not only a good introduction for graduate students and advanced undergraduates to the rich field of matrix analysis, but is an important resource for matrix computation experts who develop and analyze matrix algorithms. To a lesser extent, it is also a valuable resource for a scientist or engineer whose work involves matrix computation. The first edition of this book [1] was published in 1985, and has become a standard reference for the field. That book had eight chapters plus a "chapter 0." This second edition also has eight chapters, with some of the titles changed and with chapter 0 greatly expanded. In all of the cases, the title change signifies new material. Although some topics have been moved, most of the changes are expansions. The book now has more than 1,100 exercises, making it an excellent for use in the classroom or for independent study. The first chapter, "Eigenvalues, Eigenvectors, and Similarity," introduces the eigenvalue problem with the related concepts of characteristic polynomials, algebraic and geometric multiplicity, and left and right eigenvectors. The second chapter, "Unitary Similarity and Unitary Equivalence," has changed considerably from the first edition. Its introduction of the QR decomposition defines Householder and Givens transformations, and the decomposition theorem is proved with Householder transformations instead of classical Gram-Schmidt as in the first edition. The singular value decomposition is introduced here, whereas it was introduced in chapter 7 in the first edition. The authors add a section on the CS decomposition, which has increased in importance since 1985. Chapter 3, "Canonical Forms for Similarity and Triangular Factorizations," gives a thorough development of the Jordan canonical form. The second edition introduces a discussion of the Weyr canonical form, an interesting and different approach to similarity reduction. It finishes with variants of triangular decomposition. Chapter 4, "Hermitian Matrices, Symmetric Matrices, and Congruences," is mostly just rewritten from the first edition. I was particularly fond of the section on complex symmetric matrices. The fifth chapter, "Norms for Vectors and Matrices," is also much like its counterpart in the first edition. A fairly comprehensive review of the useful analytic and algebraic properties of norms is given. The duality discussion in Section 5.5 is quite interesting. Chapter 6, "Location and Perturbation of Eigenvalues," is mostly about Geršgorin discs and their generalizations. The last two chapters—"Positive Definite and Semidefinite Matrices" (chapter 7) and "Positive and Nonnegative Matrices" (chapter 8)—unlike the other six, contain mostly material that is rarely found in standard linear algebra or numerical analysis texts. Chapter 7 gives a more elegant proof of the von Neumann trace theorem than the first edition. It includes good discussions of unitarily invariant norms, the Schur product theorem, and some useful inequalities for positive definite matrices. The final chapter discusses positive and nonnegative matrices, which are important for Markov modeling problems. Those readers used to numerical linear algebra texts such as Golub and Van Loan's [2] will find that this book develops concepts more formally and with more detail. The notational conventions used may require adjustment for some readers. For instance, the set of square matrices is M n and the set of rectangular matrices is M m , n rather than R n × n (for all real n × n matrices) or C m × n (for all complex m × n matrices). Another example is that the authors' notation for, and development of, matrix norms is different from what is common in the numerical linear algebra literature. However, these differences in convention are unimportant. The book is well organized, completely readable, and very enlightening. For researchers in matrix analysis, matrix computations, applied linear algebra, or computational science, this second edition is a valuable book. More reviews about this item: Amazon Online Computing Reviews Service