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Linear algebra and matrices. Topics for a second course. (English)
Pure and Applied Undergraduate Texts 24. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1852-6/hbk). xv, 317~p. (2015).
This book is based on two upper level mathematics courses taught by the author: a second course in linear algebra and a course in combinatorial matrix theory. The book also covers the material of a typical linear algebra course, albeit in a terse, rigorous, theoretical way. Systems of linear equations and their solution are, for example, not included, nor are questions of numerical linear algebra dealt with. From Chapter 4 onwards, it deals with the core topics of a second course in linear algebra: the Jordan canonical form, normal matrices and the spectral theorem, Hermitian matrices, the Perron-Frobenius theorem. A valuable feature of the Jordan canonical form theory is a discussion of the various approaches and their different viewpoints. The approach chosen by the author is based on a theorem of Sylvester, the Schur triangularization theorem and a final step based on a lecture given by {\it P. Halmos} [Am. Math. Mon. 100, No. 10, 942‒944 (1993; Zbl 0792.01036)] in Pensacola. It also includes a discussion of the Weyr characteristic and normal form. For the Perron-Frobenius theorem, the author follows Wieland’s approach and uses directed graphs to deal with imprimitive matrices. The chapter headings are: 1. Preliminaries, 2. Inner product spaces and orthogonality, 3. Eigenvalues, eigenvectors, diagonalization, triangularization, 4. The Jordan and Weyr canonical forms, 5. Unitary similarity and normal spaces, 6. Hermitian matrices, 7. Vector and matrix norms, 8. Some matrix factorizations, 9. Field of values, 10. Simultaneous triangularization (including McCoy’s theorem about Property P and the Motzkin-Taussky theorem about Hermitian matrices with Property L), 11. Circulant and block cycle matrices, 12. Matrices of zeros and ones, 13. Block designs (including a proof of the Bruck-Ryser-Chowla theorem using matrix theory and elementary number theory), 14. Hadamard matrices, 15. Graphs, 16. Directed graphs, 17. Nonnegative matrices, 18. Error-correcting codes, 19. Linear dynamical systems. Each chapter ends with a set of interesting exercises extending the theory or filling in gaps. This book stands out from the legions of linear algebra books in that it deals with the theory of linear algebra and matrices with the intention of showing the beauty of the subject, in particular with respect to the proofs, as well as the natural role played by matrices in combinatorics and other fields. In this way, it makes a substantial contribution to a deeper and fuller insight and understanding of the subject and will appeal to anyone wanting more than just a working knowledge of linear algebra.
Reviewer: Rabe von Randow (Bonn)
Classification: H65
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