Matrix theory (Q2840668)

Matrices are ubiquitous objects in mathematics since the middle of the 19th century, meanwhile penetrating almost all branches of contemporary mathematics and their applications in other sciences. Apart from being one of the universal tools in modern sciences, matrix theory itself is a classical topic of algebra that still continues to be a very active area of current mathematical research, thereby exuberating with always new discoveries, insights and applications.NEWLINENEWLINE Since the publication of \textit{F. R. Gantmacher's} two-volume standard text [Applications of the theory of matrices. New York-London: Interscience Publishers (1959; Zbl 0085.01001); Matrizenrechnung. Teil 2: Spezielle Fragen und Anwendungen. (Hochschulbücher für Mathematik. Bd. 37.) Berlin: VEB Deutscher Verlag der Wissenschaften (1959; Zbl 0085.00904)] more than half a century has gone by, while the theory of matrices has made tremendous progress in the meantime.NEWLINENEWLINE The book under review presents some modern aspects of matrix theory. Based on lecture notes for graduate courses in modern matrix theory which the author has taught repeatedly at Chinese universities, this book provides a concise treatment of various topical aspects of the theory of matrices. Being primarily intended for the use as a text for graduate or advanced undergraduate level courses, the book can also serve as a guide through the more recent research literature in the field, as a source for self-study, or as a reference for active researchers. As for the prerequisites, the reader is assumed to have acquired some familiarity with the basics of matrix calculus over the field of real or complex numbers, perhaps from a first course in linear algebra.NEWLINENEWLINE The material covered in the present book is organized in ten chapters, each of which is subdivided into several sections.NEWLINENEWLINE Chapter 1 is titled ``Preliminaries'' and presents various concepts and results that will be used in the sequel. The sections of this introductory chapter briefly deal with fundamental topics such as classes of special matrices, eigenvalues, the spectral mapping theorem, matrix decompositions, the numerical range, companion matrices, generalized inverses, Schur complements, systems of linear inequalities, orthogonal projections, reducing subspaces, and others. The last section of this chapter gives hints for further reading concerning books and journals about matrices. Chapter 2 discusses tensor products of matrices, linear matrix equations, the Frobenius-König theorem, and the basic properties of compound matrices. Chapter 3 turns to Hermitian matrices, stochastic matrices, and the majorization relation. Eigenvalues of Hermitian matrices, majorization and doubly stochastic matrices, and inequalities for positive semi-definite matrices are the main themes in this context. Chapter 4 is concerned with singular values and unitarily invariant norms of matrices, thereby touching upon various classical and recent results in this direction. Chapter 5 is devoted to the perturbation theory of matrices, with a special view toward the perturbation of eigenvalues of normal matrices and polar decompositions. Chapter 6 analyzes the various special properties of nonnegative real matrices, including the essentials of Perron-Frobenius theory, the basic results on primitive and imprimitive matrices, the relation between matrices and digraphs, and two more recent theorems about positive matrices. The problem of completions of partial matrices is addressed in Chapter 7, where the relevant fundamental theorems of \textit{S. Friedland} [Isr. J. Math. 11, 184--189 (1972; Zbl 0252.15004)]; \textit{H. K. Farahat} and \textit{W. Ledermann} [Proc. Edinb. Math. Soc., II. Ser. 11, 143--146 (1959; Zbl 0090.01502)]; \textit{S. Parrott} [(1978)] and \textit{R. Grone}, \textit{C. R. Johnson}, \textit{E. M. Sá} and \textit{H. Wolkowicz} [Linear Algebra Appl. 58, 109--124 (1984; Zbl 0547.15011)] are discussed.NEWLINENEWLINE Chapter 8 studies the sign patterns of real matrices, the significance of which becomes particularly manifest in qualitative economics.NEWLINENEWLINE The main result discussed in this chapter is the theorem of \textit{M. Fiedler} and \textit{R. Grone} [Linear Algebra Appl. 40, 237--245 (1981; Zbl 0464.15009)] on characterizations of inverse-positive matrices. In Chapter 9, the author presents a number of miscellaneous topics that are also of interest and importance. Namely, he discusses results on the similarity of real matrices via complex matrices, inverses of band matrices, norm bounds for commutator matrices, the converse of the so-called ``diagonal dominance theorem'', the shape of the numerical range, J. S. Frame's simple recursion algorithm for inverting a matrix, canonical forms of matrices with respect to similarity, and an extremal sparsity property of the Jordan canonical form, respectively. The latter topic contains a fairly recent research result of \textit{R. A. Brualdi}, \textit{P. Pei} and the author [Linear Algebra Appl. 429, No. 10, 2367--2372 (2008; Zbl 1156.15003)], the proof of which is also given in this chapter. Chapter 10, the final part of the book, illustrates various applications of matrix calculus in other branches of mathematics, including combinatorics, number theory, algebra, convex geometry, and complex analysis. In this context, classical results are proved using methods from matrix theory as developed in the foregoing chapters.NEWLINENEWLINE There is a further section at the end of the book, in which the author describes the history and current state of some famous, yet still unsolved problems in matrix theory. These problems concern the existence of Hadamard matrices, the characterization of the (complex) eigenvalues of nonnegative matrices, the permanental dominance conjecture for positive semi-definite matrices, the so-called ``\(2n\) conjecture on spectrally arbitrary sign patterns'', and further more recent conjectures.NEWLINENEWLINE Each chapter of the book ends with a long list of related exercises. As the author points out, most of the exercises are taken from research papers, and therefore they are of considerable depth and difficulty. However, many concrete hints for further reading help the reader to tackle these challenging problems.NEWLINENEWLINE All in all, the present graduate textbook covers a wide spectrum of classical and contemporary matrix theory. The author discusses a number of topics that appear here for the first time in a textbook, and he enriches some classical topics with essentially new approaches and results. This and the discussion of unsolved problems in matrix theory leads the reader directly to the forefront of current research in the field, thereby providing a highly topical and useful addition to the standard texts on matrices. Besides, the book may be considered as an excellent guide to the relevant research literature in this still very active area of mathematics.