Positive Definite Matrices

Positive Definite Matrices

Positive Definite Matrices

Positive Definite Matrices

Excerpt

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 . . .

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