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Matrix Theory

Fuzhen Zhang
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a thoroughly modern introduction to matrix analysis, aimed at upper-division undergraduates with a modest background in linear algebra.  It features many short and clever proofs, often making use of partitioned matrices and an enormous number of non-drill exercises (without hints or answers).
This is a theory and proofs book; it does not cover applications at all and does not mention anything about numerical work or algorithms (except Gersgorin’s Theorem). It covers a number of unusual topics towards the end of the work, such as majorants and minorants and matrix inequalities.  The book generally takes an abstract-algebra approach, concentrating on characterizations of the various types of matrices (the champion is on pp. 293–295, where there are 37 equivalent criteria for a matrix to be normal).  The Jordan canonical form is developed very early and is used in many of the proofs. Many proofs use continuity arguments. Although it is a matrix book, we usually don’t see explicit matrices but work at a higher level of abstraction, developing our theorems from other theorems about matrices rather than by calculation. This blurs the distinction between matrix theory and linear algebra (a good thing, in my opinion). We are here interested in linear transformations having particular properties and in the structure of linear transformations, rather than in the structure of linear spaces. The flavor of this book is similar to that of Axler’s Linear Algebra Done Right.
The standard work in this field is probably Horn and Johnson’s Matrix Analysis. The two books are similar in approach. Horn and Johnson is aimed higher and assumes the reader has already been through an undergraduate linear algebra course (they review everything in Chapter 0), and they cover more topics in more depth. They also have an enormous number of non-drill exercises but do provide hints or answers for many of them.
The big weakness of the present work is the references, which although plentiful (11 pages of books and papers) are not referenced anywhere from the body of the work. There’s no way to trace a particular result to its source.
Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal website is His mathematical interests are number theory and classical analysis.