Preface
The theory of positive definite matrices, positive definite functions, and positive linear maps is rich in content. It offers many beautiful theorems that are simple and yet striking in their formulation, uncomplicated and yet ingenious in their proof, diverse as well as powerful in their application. The aim of this book is to present some of these results with a minimum of prerequisite preparation.
The seed of this book lies in a cycle of lectures I was invited to give at the Centro de Estruturas Lineares e Combinatórias (CELC) of the University of Lisbon in the summer of 2001. My audience was made up of seasoned mathematicians with a distinguished record of research in linear and multilinear algebra, combinatorics, group theory, and number theory. The aim of the lectures was to draw their attention to some results and methods used by analysts. A preliminary draft of the first four chapters was circulated as lecture notes at that time. Chapter 5 was added when I gave another set of lectures at the CELC in 2003.
Because of this genesis, the book is oriented towards those interested in linear algebra and matrix analysis. In some ways it supplements my earlier book Matrix Analysis (Springer, Graduate Texts in Mathematics, Volume 169). However, it can be read independently of that book. The usual graduate-level preparation in analysis, functional analysis, and linear algebra provides adequate background needed for reading this book.
Chapter 1 contains some basic ideas used throughout the book. Among other things it introduces the reader to some arguments involving 2 × 2 block matrices. These have been used to striking, almost magical, effect by T. Ando, M.-D. Choi, and other masters of the subject and the reader will see some of that in later parts of the book.
Chapters 2 and 3 are devoted to the study of positive and completely positive maps with special emphasis on their use in proving matrix inequalities. Most of this material is very well known to those who study C∗-algebras, and it ought to be better known to workers in linear algebra. In the book, as in my Lisbon lectures, I have avoided the technical difficulties of the theory of operator algebras by staying in finite-dimensional spaces. Thus some of the major theorems of the subject are presented in their toy versions. This is good enough for the purposes of matrix analysis and also of the currently popular area of quantum information theory. Quantum communication channels, at present, are thought of as completely positive trace-preserving linear maps on matrix algebras and many problems of the subject are phrased in terms of block matrices.
In Chapter 4 we discuss means of two positive definite matrices with special emphasis on the geometric mean. Among spectacular applications of these ideas we include proofs of some theorems on matrix convex functions, and of two of the most famous theorems on quantum mechanical entropy.
Chapter 5 gives a quick introduction to positive definite functions on the real line. Many examples of such functions are constructed using elementary arguments and then used to derive matrix inequalities. Again, a special emphasis has been placed on various means of matrices. Many of the results presented are drawn from recent research work.
Chapter 6 is, perhaps, somewhat unusual. It presents some standard and important theorems of Riemannian geometry as seen from the perspective of matrix analysis. Positive definite matrices constitute a Riemannian manifold of nonpositive curvature, a much-studied object in differential geometry. After explaining the basic ideas in a language more familiar to analysts we show how these are used to define geometric means of more than two matrices. Such a definition has been elusive for long and only recently some progress has been made. It leads to some intriguing questions for both the analyst and the geometer.
This is neither an encyclopedia nor a compendium of all that is known about positive definite matrices. It is possible to use this book for a one semester topics course at the graduate level. Several exercises of varying difficulty are included and some research problems are mentioned. Each chapter ends with a section called “Notes and References”. Again, these are written to inform certain groups of readers, and are not intended to be scholarly commentaries.
The phrase positive matrix has been used all through the book to mean a positive semidefinite, or a positive definite, matrix. No confusion should be caused by this. Occasionally I refer to my book Matrix Analysis. Most often this is done to recall some standard result. Sometimes I do it to make a tangential point that may be ignored without losing anything of the subsequent discussion. In each case a reference like “MA, page xx” or “See Section x.y.z of MA” points to the relevant page or section of Matrix Analysis.
Over the past 25 years I have learnt much from several colleagues and friends. I was a research associate of W. B. Arveson at Berkeley in 1979–80, of C. Davis and M.-D. Choi at Toronto in 1983, and of T. Ando at Sapporo in 1985. This experience has greatly influenced my work and my thinking and I hope some of it is reflected in this book. I have had a much longer, and a more constant, association with K. R. Parthasarathy. Chapter 5 of the book is based on work I did with him and the understanding I obtained during the process. Likewise Chapter 6 draws on the efforts J.A.R. Holbrook and I together made to penetrate the mysteries of territory not familiar to us.
D. Drissi, L. Elsner, R. Horn, F. Kittaneh, K. B. Sinha, and X. Zhan have been among my frequent collaborators and correspondents and have generously shared their ideas and insights with me. F. Hiai and H. Kosaki have often sent me their papers before publication, commented on my work, and clarified many issues about which I have written here. In particular, Chapter 5 contains many of their ideas.
My visits to Lisbon were initiated and organized by J. A. Dias da Silva and F. C. Silva. I was given a well-appointed office, a good library, and a comfortable apartment—all within 20 meters of each other, a faithful and devoted audience for my lectures, and a cheerful and competent secretary to type my notes. In these circumstances it would have been extraordinarily slothful not to produce a book.
The hard work and good cheer of Fernanda Proença at the CELC were continued by Anil Shukla at the Indian Statistical Institute, Delhi. Between the two of them several drafts of the book have been processed over a period of five years.
Short and long lists of minor and major mistakes in the evolving manuscript were provided by helpful colleagues: they include J. S. Aujla, J. C. Bourin, A. Dey, B. P. Duggal, T. Furuta, F. Hiai, J.A.R. Holbrook, M. Moakher, and A. I. Singh. But even their hawk eyes might have missed some bugs. I can only hope these are both few and benignant.
I am somewhat perplexed by authors who use this space to suggest that their writing activities cause acute distress to their families and to thank them for bearing it all in the cause of humanity. My wife Irpinder and son Gautam do deserve thanks, but my writing does not seem to cause them any special pain.
It is a pleasure to record my thanks to all the individuals and institutions named above.