Matrices: Theory and Applications

ISBN-13: 9781461427230 / Angielski / Miękka / 2012 / 289 str.

Denis Serre

Opis

In this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. With forty percent new material, this second edition is significantly different from the first edition. Newly added topics include: Dunford decomposition, tensor and exterior calculus, polynomial identities, regularity of eigenvalues for complex matrices, functional calculus and the Dunford-Taylor formula, numerical range, Weyl's and von Neumann's inequalities, and Jacobi method with random choice. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the Ecole Normale Superieure de Lyon.

Recenzje

From the reviews of the second edition:

"This is a first-year graduate text that will be of considerable value to students with any of a wide variety of potential research objectives. Both the proofs and the general explanations are unusually detailed...There are 165 well-designed exercises, many of which provide additional information on topics treated in the text." (Mathematical Reviews)

"The main motivation of the author is to ... obtain a textbook that is suitable for a wide range of applications and hence for the different audiences deeply interested in matrices. The main goal is certainly achieved. ... The complex theoretical language, the presentation of modern numerical techniques and the classical algebraic basics make this textbook an excellent source for graduate students in each field using matrices. It can be recommended both for use in the classroom and for independent study." (Péter Hajnal, Acta Scientiarum Mathematicarum, 71, 2005)

"This is a first-year graduate text that will be of considerable value to students with any of a wide variety of potential research objectives. Both the proofs and the general explanations are unusually complete and detailed ... . There are 165 well-designed exercises, many of which provide additional information on topics treated in the text. Full solutions are available on the author's website." (R.J. Bumcrot, Mathematical Reviews, 2003)

"Denis Serre provides a clear and concise introduction to the basic theory of matrices. ... The book combines algebra, analysis, complexity theory, and numerical analysis, and it will provide many scientists, not just mathematicians, with a useful and reliable reference. ... The book is intended for advanced undergraduate and graduate students with either applied or theoretical goals." (L'enseignement mathématique, 48:3-4, 2002)

"Serre ... offers another volume in this series of mathematics work; it is designed for graduate students or upper-level undergraduates who wish to study matrix theory or need a reference in that subject. The discussion is approached from a strongly algebraic viewpoint and is quite concise and thorough. ... There are about 170 exercises spread throughout, all theoretical in nature. Extensive bibliography. A useful resource for anyone who uses matrix theory in his or her work. Summing Up: Highly recommended." (J.H. Ellison, Choice, April 2003)

"This text offers an advanced course in matrix theory aimed at a student with a good background in analysis, providing a selection of advanced topics in matrices over the real or complex fields without attempting to be encyclopedic. ... The book contains a large number of exercises, many interesting and challenging, including extensions of results discussed in the main text and alternative proofs of some theorems." (J.D. Dixon, Zentralblatt MATH, 1011, 2003)

"It covers, at a slightly higher than basic level, all these expects, and is therefore well suited for a graduate program. ... The prose is good, there are plenty of exercises, and the book is accordingly well suited for self-study. Additionally Matrices would serve very well as a text for courses locally homeomorphic to Serre's, of which there should clearly be more." (Michael Berg, The Mathematical Association of America, December, 2010)

"Containing numerous highlights and details that cannot be found in other texts, this book can be recommended warmly for an advanced course on matrix theory." (H. Mitsch, Monatshefte für Mathematik, Vol. 165 (1), January, 2012)

"The author has added many new exercises extending the basic material and these add considerably to the value of the book. ... the author touches on many aspects ... offering a range of topics, and presenting some fascinating and perhaps not familiar parts of the subject. Even those familiar with the subject may find something new here, and a good student, willing to work hard, could find it a rewarding book for self-study." (John D. Dixon, Zentralblatt MATH, Vol. 1206, 2011)

Spis treści

Preface to the Second Edition.- Preface to the First Edition.- List of Symbols.- 1 Elementary Linear and Multilinear Algebra.- 2 What Are Matrices.- 3 Square Matrices.- 4 Tensor and Exterior Products.- 5 Matrices with Real or Complex Entries.- 6 Hermitian Matrices.- 7 Norms.- 8 Nonnegative Matrices.- 9 Matrices with Entries in a Principal Ideal Domain; Jordan Reduction.- 10 Exponential of a Matrix, Polar Decomposition, and Classical Groups.- 11 Matrix Factorizations and Their Applications.- 12 Iterative Methods for Linear Systems.- 13 Approximation of Eigenvalues.- References.- Index of Notation.- General Index.- Cited Names.-

O autorze

Denis Serre is Professor of Mathematics at École Normale Supérieure de Lyon and a former member of the Institut Universitaire de France. He is a member of numerous editorial boards and the author of "Systems of Conservation Laws" (Cambridge University Press 2000). With S. Benzoni-Gavage, he is the co-author of "Multi-Dimensional Hyperbolic Partial Differential Equations. First Order Systems and Applications" (Oxford University Press 2007). With S. Friedlander, he has co-edited four volumes of a "Handbook of Mathematical Fluid Dynamics" (Elsevier 2002--2007). The first edition of the present book is a translation of the original French edition, "Les Matrices: Théorie et Pratique", published by Dunod (2001).

Blurb

In this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. With forty percent new material, this second edition is significantly different from the first edition. Newly added topics include: • Dunford decomposition, • tensor and exterior calculus, polynomial identities, • regularity of eigenvalues for complex matrices, • functional calculus and the Dunford–Taylor formula, • numerical range, • Weyl's and von Neumann’s inequalities, and • Jacobi method with random choice. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the École Normale Supérieure de Lyon.