Preface

Summary

This book is aimed at first year graduate students as well as research workers with a background in linear algebra. The theory of nonnegative matrices is unfolded in the book using tools from optimization, inequalities and combinatorics. The topics and applications are carefully chosen to convey the excitement and variety that nonnegative matrices have to offer. Some of the applications also illustrate the depth and the mathematical elegance of the theory of nonnegative matrices. The treatment is rigorous and almost all the results are completely proved. While about half of the material in the book presents many topics in a novel fashion, the remaining portion reports many new results in matrix theory for the first time in a book form. Although the only prerequisite is a first course in linear algebra and advanced calculus, familiarity with linear programming and statistics will be helpful in appreciating some sections.

To give some examples, the Perron-Frobenius Theorem and many of its consequences are derived using the theory of matrix games where all rows and columns are essential for optimal play. The chapter on conditionally positive definite matrices and distance matrices has several new results appearing for the first time in a book. A transparent proof of the Alexandroff inequality for mixed discriminants is presented and a characterization of graphs giving rise to a finite Coxeter group is given in the chapter on combinatorial theory.