This book will definitely be useful for specialists in algebra, that is, professional mathematicians. Although published in a series on applied mathematics, for an applied scientist it may be mainly a source of very general inspiration or an incentive to broaden scientific horizons. The book is very demanding for a technical reader and does not focus on engineering applications. Nevertheless, the style is very elegant and the author provides examples and exercises, as well as useful references to related literature.

Concerning the contents, chapter 1 gives the notion of positive definite and positive semidefinite matrices and the related fundamental theorems. The issues are presented in general with elements coming from Hilbert spaces. Also, the notion of block matrices, very important throughout the book, is elaborated. Then, in chapter 2, Bhatia discusses the notion of linear maps (a multi-dimensional generalization of a linear functional) and their relationship to the theory of positive (semi)definite matrices. This concept is extended with the notion of complete positive maps in chapter 3. The well-known Schwartz inequality is derived in this context. The next chapter may be closest to potential applications, as the notions of various means (arithmetic, geometric, and so on) are generalized to the matrix-related context. Chapter 5 deals with a class of positive definite functions. This part requires familiarity with Fourier analysis. The last chapter deals with geometrical interpretation and properties of positive definite matrices.

As a source of inspiration, this book may be read during PhD seminars or by specialists with an extensive mathematical background. I would recommend it to professionals in optimization, image processing, machine learning, and signal or coding theory, since they should be well acquainted with many of the notions presented.

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