A survey of matrix theory and matrix inequalities Marcus, Marvin

This book presents an enormous amount of information in a concise and accessible format. Beginning with the assumption that the reader has never seen a matrix before, the authors go on to provide a survey of a substantial part of the field, including many areas of modern research interest. Part one of the book covers matrix theory, compound and induced matrices, quadratic relations, incidence matrices and generalizations of commutativity. Part two begins with a survey of elementary properties of convex sets and polyhedra and presents a proof of the Birkhoff theorem on doubly stochastic matrices. This is followed by a discussion of the properties of convex function and a list of classical inequalities. This material is then combined to yield many of the interesting matrix inequalities of Weyl, Fan, Kantorovich and others. Th treatment proofs are included. This chapter contains an account of the classical Perron-Frobenius-Wielandt theory of indecomposable non negative matrices and ends with some important results on stochastic matrices. Part Three is concerned with a variety of results on the localization of the characteristic of a matrix in terms of simple functions of its entries of entries of related matrix. The presentation is essentially in historical order, and out of the vast number of results in this field the authors have culled those that seemed most interesting or useful. Readers will find many of the proofs of classical theorems and substantial number of proofs of results in contemporary research literature.