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Matrices: Algebra, Analysis and Applications

Shmuel Friedland
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
, on

This book on advanced topics in linear algebra can serve as a reference for professionals, or as a text for a graduate course for those aspiring to be professionals. This would be, however, quite a sophisticated course. This book is written by an expert, and is not intended for neophytes. The author’s writing style is terse: what needs to be said, is said, but there is very little in the way of chatty exposition or hand-holding. It is, for example, considerably more demanding than the similarly-titled Matrix Analysis by Horn and Johnson.

There are seven chapters with a reasonable degree of independence. The first chapter introduces a variety of different kinds of integral domains and discusses modules and matrices over them. The second chapter discusses canonical forms; the third, functions of matrices; the fourth, inner product spaces. Chapter 5 addresses multilinear algebra (tensor products, exterior powers, symmetric powers, etc.). Chapter 6, which uses concepts from graph theory, discusses non-negative matrices. A final chapter (“Various Topics”) discusses a variety of topics, including norms and spectral radius and the inverse eigenvalue problem for non-negative matrices. The choice of topics, according to the back-cover blurb, “is very personal and reflects the subjects that the author was actively working on in the last 40 years.”

Although many of the topics mentioned above may seem standard, the exposition here is definitely not: these topics are addressed at a level beyond that found in most of the standard textbook literature. Indeed, each chapter contains material that is, in Friedland’s words, “known to the experts”, as well as material that is new. Consistent with the back-cover blurb quoted in the previous paragraph, the fifteen-page bibliography at the end of the text lists about 30 research papers naming Friedland as a co-author.

The prerequisites for reading this text are not specified, but they are considerable. Obviously a solid background in linear algebra is essential. Moreover, although the author defines terms like “ring” from scratch, it seems clear that prior exposure to abstract algebra is also necessary. Background in analysis and point-set topology is also necessary: as early as page 2, for example, the author invokes analytic functions in one and several complex variables; complex analysis is, in fact, used throughout the text. On occasion, proofs are supplied that use sheaves.

Conclusion: Not a book for the faint-hearted, but people who do, or who plan to do, research in the topics in linear algebra that are covered here, will undoubtedly find this to be a very valuable book.

Mark Hunacek ( teaches mathematics at Iowa State University. 

  • Domains, Modules and Matrices
  • Canonical Forms for Similarity
  • Functions of Matrices and Analytic Similarity
  • Inner Product Spaces
  • Elements of Multilinear Algebra
  • Non-Negative Matrices
  • Various Topics