xii Preface such as Gantmacher [Gan59], Halmos [Hal87], Mal’cev [Mal63], Hoffman and Kunze [HK71], Bellman [Bell70]. More recently, we have the excellent book by Horn and Johnson [HJ85], now expanded in the second edition [HJ13]. But, much as I admire these books, they weren’t quite what I wanted for my courses aimed at upper level undergraduates. In some cases I wanted different topics, in others I wanted a different approach to the proofs. So perhaps there is room for another linear algebra book. This book is not addressed to experts in the field nor to the would-be matrix theory specialist. My hope is that it will be useful for those (both students and professionals) who have taken a first linear algebra course but need to know more. A typical linear algebra course taken by mathematics, science, engineering, and economics majors gets through systems of linear equations, vector spaces, inner products, determinants, eigenvalues and eigenvectors, and maybe diagonalization of symmetric matrices. But often in mathematics and applications, one needs more advanced results, such as the spectral theorem, Jordan canonical form, the singular value decomposition. Linear algebra plays a key role in so many parts of mathematics: linear dif- ferential equations, Fourier series, group representations, Lie algebras, functional analysis, multivariate statistics, etc. If the two courses mentioned in the first para- graph are the parents of this book, the aunts and uncles are courses I taught in differential equations, partial differential equations, number theory, abstract alge- bra, error-correcting codes, functional analysis, and mathematical modeling, all of which used linear algebra in a central way. Thank you to all the chairs of the Swarthmore College Mathematics Department for letting me teach such a variety of courses. In the case of the modeling course, a special thank you to Thomas Hunter for gently persuading me to take this on when no one else was eager to do it. One more disclaimer: this is not a book on numerical linear algebra with discus- sion of efficient and stable algorithms for actually computing eigenvalues, normal forms, etc. I occasionally comment on the issues involved and the desirability of working with unitary change of basis, if possible, but my acquaintance with this side of the subject is very limited. This book does not contain new results. It may contain some proofs not typ- ically seen or perhaps not readily available elsewhere. In the proof of the Jordan canonical form, the argument for the last nitty-gritty part comes from a lecture Halmos gave at the 1993 ILAS conference. He explained that he had never been satisfied with the argument in his book and always thought there should be a more conceptual approach and that he had finally found it. My apologies if the account I give here is more complicated than necessary—my notes from the talk had a sketch of a proof and then I needed to fill in details. A former colleague, Jim Wiseman, showed me the shear argument with the spherical coordinates used in the proof that the numerical range of a 2 × 2 matrix is an ellipse. From Hans Schneider I learned about the connection between the eigenvalue structure for an irreducible nonnegative matrix and the cycle lengths of its directed graph, and the proof in this book starts with the graph and uses it to obtain the usual result about the eigenvalues.
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