The ranking/rating process is deeply embedded in human nature and our everyday lives. Whenever we need to make a decision that involves more than one alternative, such as buying a product, voting for a candidate, presenting a list of web pages after a web search, or selecting a university for our studies, we have to arrange a group of items in order of importance. We assign a rating to each item, which will finally provide us with a ranked list of the items.

This book presents fundamental ideas behind mathematical rating systems. It confines itself to those rating/ranking methods that are based on linear algebra. It is written by two scientists, both well-known experts in the field of linear algebra.

The book consists of 18 chapters, 15 of which describe ranking and rating methods. The book is full of instructive examples, making it ideal for self-study, even though a decent background in linear algebra is required to appreciate the beauty of the book.

One weakness of this book is the lack of a chapter surveying other rating/ranking methods that don’t rely on matrix computations.

Even though the book is written with a clear focus on ranking sports teams, its actual audience is far bigger than that. It potentially includes undergraduate and postgraduate computer science and engineering students studying data mining (recommendations), information retrieval, and social networks. It is also useful to practitioners interested in politics, chess players, and social choice theorists.

Overall, it is a very valuable book that compiles these mathematical formalisms into a single source of study, thus filling a gap in the relevant literature.

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