Note to the Reader The basic prerequisite for this book is familiarity with the linear algebra and matrix theory typically covered in a first course: systems of linear equations, linear trans- formations, inner product spaces, eigenvalues, determinants, etc. Some sections and proofs use basic concepts from algebra and analysis (quotient spaces, cosets, convergence, open and closed sets, etc.) readers unfamiliar with these concepts should just skip those parts. For those using this book as a text for a course, there are various options. The author has used an earlier form of some chapters for two different courses. One was a linear algebra course, covering material from Chapters 2, 3, 4, 5, 6, and 17. The other was a course in combinatorial matrix theory, focused on material from Chapters 11 through 17. Material needed from Chapters 2, 3, and 4 was mentioned as needed. It is my hope that users will find a variety of choices that suit them. The complete proof of the Jordan canonical form can eat up a lot of class time—those who prefer to include more of the other topics should probably just outline, or skip, the proof. When the Jordan form is needed later, it is sufficient to know what it is. I also hope that some chapters are useful on their own as introductions to particular topics: block designs, Hadamard matrices, error-correcting codes, linear dynamical systems, Markov chains. Many well-known results are typically mentioned without reference to the orig- inal sources in some cases, it is not so clear what the original source is. In a field like linear algebra—used as a basic tool for so much of mathematics—important results may be discovered and rediscovered several times by different people and in different contexts. I have made some attempt to cite some of the original papers but make no claim to have been complete or consistent in this endeavor. For more information on older original sources, I recommend MacDuffee’s book The Theory of Matrices [MacD33]. I have also found the “Notes and References” section at the end of each chapter of [StewSun90] very helpful. Despite several rounds of error detection and correction, I have surely missed some. I apologize for these and hope readers will let me know where they are. xv
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