A Survey of Matrix Theory and Matrix Inequalities

Each section starts with a set of necessary definitions. Then, theorems are stated without proof---therefore, the it can also be used as an exercise book. One can go to a section of interest and try to prove some of the theorems listed there.

I have just read part I of the book (table of contents below), and I like it very much. So many information, on so few pages. Everything is threated in FULL generality, and proves are included for the main theorems. This book has also an excellent encyclopedia function, as it contains the main properties of anything that one can think of. Beware: I have some mathematical background, and if you fully want to appreciate this book you will need this. (don't buy this book if this would be your first confrontation with matrices).Table of contents:I Survey of matrix theory 1 Introductory concepts 2 Numbers associated with matrices 3 Linear equations and canonical forms 4 Special classes of matrices, commutativity 5 CongruenceII Convexity and matrices 1 Convex sets 2 Convex functions 3 Classical Inequalities4 Convex functions and matrix inequalities. 5 Nonnegative matricesIII Localization of characteristic roots 1 Bounds for characteristic roots 2 Regions containing characteristic roots of a general matrix 3 Characterstic roots of a general matrix 4 The spread of a matrix 5 The field of values of a matrix

This book is not really a review of matrix theory; the focus is on inequalities, mainly for PD matrices, convex functions and eigenvalue problems. The book tries to present a general framework, a-la Hardy, but is not quite as successful. Nevertheless, if you need to find that elusive inequality that you saw in undergraduate, this book will likely contain it. This book should be on every mathematical researcher's shelf.