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Positive definite matrices. (English) Zbl 1133.15017
Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12918-1/hbk). ix, 254 p. (2007).
This is an outstanding book. Its exposition is both concise and leisurely at the same time. Some theorems are given more than one proof. Exercises can be found scattered throughout the text and in problems sections at the end of each chapter. Many exercises have been provided with hints. Each chapter ends with an extensive section giving attributions and additional references. There are many wonderful insights and some first rate exposition of important ideas not easily extracted from other sources. The book is a valuable reference book as well a source book for graduate students and researchers in Linear Algebra, Matrix Analysis and Operator Theory. It contains several open problems.
Chapter 1 (Positive matrices) is a review of some of the basic properties of positive definite matrices used throughout the book.
In Chapter 2 (Positive linear maps) linear maps \(\Phi \) from the algebra of \(m\times m\) matrices to the algebra of \(n\times n\) matrices which maps positive definite matrices into positive definite matrices are studied. Such a map \(\Phi \) is said to be positive. It is said to be unital if it maps identity matrix into identity matrix. Apart from basic properties for a unital positive linear map many interesting results due to Choi, Kadison, Krein Extension Theorem and Russo-Dye Theorem are included.
Chapter 3 (Completely positive maps) is devoted to the study of completely positive maps. The author study the basic properties of this special class of positive maps. He then proves Schwarz type inequalities for this class. The general results on completely positive maps are used to study matrix norms. The fundamental results due to Ando, Averson, Choi, Kraus and Stinespring are proved in this chapter.
Chapter 4 (Matrix means) is devoted to the study of matrix means. The author define a geometric mean of two positive definite matrices and study its properties along with those of the arithmetic mean and the harmonic mean. The author discuss various different definitions of geometric mean. He uses matrix means to prove some theorems on operator monotonicity and operator convexity. These theorems are then used to derive some important properties of the quantum entropy.
In Chapter 5 (Positive Definite Functions) basic properties of positive definite functions are studied. These results have been used to prove results about positive definite matrices. Elegant proofs of Herglotz’s Theorem and Bochner’s Theorem are given.
The set of \(n\times n\) positive definite matrices is a differentiable manifold with a natural Riemannian structure. The geometry of this manifold is intimately connected with some matrix inequalities. This connection has been demonstrated in Chapter 6 (Geometry of positive matrices).

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
26E60 Means
15A45 Miscellaneous inequalities involving matrices
47A64 Operator means involving linear operators, shorted linear operators, etc.
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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