Matrices

1. Front Matter

   Pages 1-1

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2. Elementary Linear and Multilinear Algebra

   • Denis Serre

   Pages 1-14

3. What Are Matrices

   • Denis Serre

   Pages 15-30

4. Square Matrices

   • Denis Serre

   Pages 31-68

5. Tensor and Exterior Products

   • Denis Serre

   Pages 69-81

6. Matrices with Real or Complex Entries

   • Denis Serre

   Pages 83-108

7. Hermitian Matrices

   • Denis Serre

   Pages 109-125

8. Norms

   • Denis Serre

   Pages 127-148

9. Nonnegative Matrices

   • Denis Serre

   Pages 149-162

10. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction

    • Denis Serre

    Pages 163-181

11. Exponential of a Matrix, Polar Decomposition, and Classical Groups

    • Denis Serre

    Pages 183-205

12. Matrix Factorizations and Their Applications

    • Denis Serre

    Pages 207-223

13. Iterative Methods for Linear Systems

    • Denis Serre

    Pages 225-245

14. Approximation of Eigenvalues

    • Denis Serre

    Pages 247-275

15. Back Matter

    Pages 271-271

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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.

• Approximation of eigenvalues
• Determinant, Pfaffian
• Eigenvalue
• Eigenvalues, localization
• Exponential, clas
• Functional calculus
• Invariant theory
• Matrix
• Multilinear Algebra
• Normal form
• Numerical range
• Singular values
• Tensor and exterior calculus
• linear algebra
• numerical analysis
• matrix theory

• Unité de Mathématiques Pures et Appliqué, École Normale Supérieure de Lyon Unité de Mathématiques Pures et Appliqué, Lyon Cedex 07, France

  Denis Serre

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).

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