**Graduate Studies in Mathematics**

Volume: 147;
2013;
264 pp;
Hardcover

MSC: Primary 15; 05; 47;

Print ISBN: 978-0-8218-9491-0

Product Code: GSM/147

List Price: $65.00

AMS Member Price: $52.00

MAA Member Price: $58.50

**Electronic ISBN: 978-1-4704-0945-6
Product Code: GSM/147.E**

List Price: $65.00

AMS Member Price: $52.00

MAA Member Price: $58.50

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#### Supplemental Materials

# Matrix Theory

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*Xingzhi Zhan*

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.

#### Readership

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

#### Reviews & Endorsements

[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

#### Table of Contents

# Table of Contents

## Matrix Theory

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Preliminaries 112 free
- Tensor products and compound matrices 3546
- Hermitian matrices and majorization 5162
- Singular values and unitarily invariant norms 7788
- Perturbation of matrices 103114
- Nonnegative matrices 119130
- Completion of partial matrices 149160
- Sign patterns 165176
- Miscellaneous topics 181192
- Applications of matrices 213224
- Unsolved problems 227238
- Bibliography 237248
- Notation 249260
- Index 251262 free
- Back Cover Back Cover1265