Matrix theory has developed rapidly in the last few decades because of its wide range of applications and many connections to different subjects. It is difficult to write an advanced level text covering all aspects of the subject nowadays. A compromise is to focus on a certain theme and some selected topics. Such an approach has been used by other authors, such as Horn and Johnson . The book Matrix Analysis by Bhatia gives another nice example for this approach.
As stated in the preface, the purpose of the book is to present a systematic treatment of the part of matrix analysis that is functional analytic in spirit. The author also mentions that a subtitle of the book could be "Matrix Inequalities" and expects that a reader who works through the book should become proficient in the art of deriving matrix inequalities. With these specific objectives in mind, he has made a careful selection and a nice arrangement of topics in the book.
The first two chapters are mainly for preparation. Basic background is presented with an emphasis on topics that are useful for future discussion. For example, sections in Chapter I, "A Review of Linear Algebra," are devoted to tensor products and symmetry classes so that the results can be used conveniently in later chapters to derive matrix inequalities. Actually, using multilinear techniques is an efficient approach to deriving matrix inequalities, as suggested by authors such as Marvin Marcus . Another useful technique for obtaining matrix inequalities is the theory of majorization (see and [1] for a detailed treatment of this subject). In Chapter II, "Majorization and Doubly Stochastic Matrices," a concise discussion of the subject is given.