Nonnegative Matrices and Applications (Encyclopedia of Mathematics and its Applications, Series Number 64)

This book presents an integrated treatment of the theory of nonnegative matrices, emphasizing connections with the themes of game theory, combinatorics, inequalities, optimization, and mathematical economics. Some related classes of positive matrices such as positive semidefinite matrices, M-matrices, P-matrices, and distance matrices are also discussed, but the main emphasis is on entrywise nonnegative matrices. The book begins with the basics of the subject, such as the Perron-Frobenius Theorem. Only a minimal background in linear algebra is assumed, although familiarity with linear programming and statistics will be helpful in following some sections. Each of the later chapters is devoted to an area of applications, including doubly stochastic matrices (price fixing, scheduling, and the fair division problem), combinatorial matroids, and economics. These applications have been carefully chosen both for their elegant mathematical content and for their accessibility. The treatment is rigorous and almost all results are proved completely. About half of the material in the book presents standard topics in a novel fashion, the remaining portion reports many new results in matrix theory for the first time in a book form. This book provides an integrated treatment of the theory of nonnegative matrices and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimization and mathematical economics. The authors have chosen the wide variety of applications, which include price fixing, scheduling, and the fair division problem, both for their elegant mathematical content and for their accessibility to students with minimal preparation. They present many new results in matrix theory for the first time in book form, while they present more standard topics in a novel fashion. The treatment is rigorous and almost all results are proved completely. These new results and applications will be of great interest to researchers in linear programming, statistics, and operations research. The minimal prerequisites also make the book accessible to first year graduate students. This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students. New results in matrix theory are presented for the first time in this study of matrices with only positive numbers, or zero, as entries. The results will be of great value to researchers in linear programming, statistics and operations research

An integrated approach for new graduate students, emphasising connections with game theory, optimisation, mathematical programming and statistics.