Preface This book began with two upper level mathematics courses I taught at Swarthmore College: a second course in linear algebra and a course in combinatorial matrix theory. In each case, I had expected to use an existing text, but then found these did not quite fit my plans for the course. Consequently, I wrote up complete notes for the classes. Since the material on nonnegative matrices belonged in both courses and required a fair amount of graph theory, it made sense to me to combine all of these chapters into one book. Additional chapters on topics not covered in those courses were added, and here is the total. I started with topics I view as core linear algebra for a second course: Jordan canonical form, normal matrices and the spectral theorem, Hermitian matrices, the Perron–Frobenius theorem. I wanted the Jordan canonical form theory to include a discussion of the Weyr characteristic and Weyr normal form. For the Perron–Frobenius theorem, I wanted to follow Wielandt’s approach in [Wiel67] and use directed graphs to deal with imprimitive matrices hence the need for a chapter on directed graphs. For the combinatorial matrix theory course, I chose some favorite topics included in Herbert Ryser’s beautiful courses at Caltech: block designs, Hadamard matrices, and elegant theorems about matrices of zeros and ones. But I also wanted the book to include McCoy’s theorem about Property P, the Motzkin–Taussky theorem about Hermitian matrices with Property L, the field of values, and other topics. In addition to linear algebra and matrix theory per se, I wanted to display linear algebra interacting with other parts of mathematics hence a brief section on Hilbert spaces with the formulas for the Fourier coefficients, the inclusion of a proof of the Bruck–Ryser–Chowla Theorem which uses matrix theory and elementary number theory, the chapter on error-correcting codes, and the introduction to linear dynamical systems. Do we need another linear algebra book? Aren’t there already dozens (perhaps hundreds) of texts written for the typical undergraduate linear algebra course? Yes, but most of these are for first courses, usually taken in the first or second year. There are fewer texts for advanced courses. There are well-known classics, xi
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