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Matrix Theory
Xingzhi Zhan, East China Normal University, Shanghai, China
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2013; 264 pp; hardcover
Volume: 147
ISBN-10: 0-8218-9491-9
ISBN-13: 978-0-8218-9491-0
List Price: US$65 Member Price: US$52
Order Code: GSM/147

Matrix theory is a classical topic of algebra that had originated, in its current form, in the middle of the 19th century. It is remarkable that for more than 150 years it continues to be an active area of research full of new discoveries and new applications.

This book presents modern perspectives of matrix theory at the level accessible to graduate students. It differs from other books on the subject in several aspects. First, the book treats certain topics that are not found in the standard textbooks, such as completion of partial matrices, sign patterns, applications of matrices in combinatorics, number theory, algebra, geometry, and polynomials. There is an appendix of unsolved problems with their history and current state. Second, there is some new material within traditional topics such as Hopf's eigenvalue bound for positive matrices with a proof, a proof of Horn's theorem on the converse of Weyl's theorem, a proof of Camion-Hoffman's theorem on the converse of the diagonal dominance theorem, and Audenaert's elegant proof of a norm inequality for commutators. Third, by using powerful tools such as the compound matrix and Gröbner bases of an ideal, much more concise and illuminating proofs are given for some previously known results. This makes it easier for the reader to gain basic knowledge in matrix theory and to learn about recent developments.

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Graduate students, research mathematicians, and engineers interested in matrix theory.

Reviews

"[I]n an orbit of some 250 pages or so [Zhan] travels from where a good undergraduate course (even in today's model) leaves off ... and then hits a host of rather marvelous themes including the inner life of Hermitian matrices and matrix perturbation theory, as well as some pretty exotic material such as the Frobenius-König Theorem and Perron-Frobenius theory. ... There are plenty of exercises to be had, and the author's goal is clearly to guide able and willing graduate students toward research in this area, which certainly possesses the attractive qualities of being both accessible ... and exciting -- it's algebra after all! I think Zhan will be successful in this enterprise: it's a very nice book indeed."

-- Michael Berg, MAA Reviews