id: 06504873
dt: b
an: 2016c.00798
au: Shapiro, Helene
ti: Linear algebra and matrices. Topics for a second course.
so: Pure and Applied Undergraduate Texts 24. Providence, RI: American
Mathematical Society (AMS) (ISBN 978-1-4704-1852-6/hbk). xv, 317~p.
(2015).
py: 2015
pu: Providence, RI: American Mathematical Society (AMS)
la: EN
cc: H65
ut: textbook; combinatorial matrix theory; Jordan canonical form; normal
matrices; spectral theorem; Hermitian matrices; Perron-Frobenius
theorem; Schur triangularization; Weyr characteristic; normal form;
directed graph; inner product space; eigenvalue; eigenvector;
diagonalization; triangularization; Weyr canonical form; unitary
similarity; normal space; vector norm; matrix norm; matrix
factorization; field of values; McCoy’s theorem; Motzkin-Taussky
theorem; circulant and block cycle matrices; block design;
Bruck-Ryser-Chowla theorem; Hadamard matrices; nonnegative matrices;
error-correcting codes; linear dynamical system
ci: Zbl 0792.01036
li:
ab: 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.
rv: Rabe von Randow (Bonn)