This book is meant as a text for a serious second course on Linear Algebra. The intended audience is beginning graduate students and advanced undergraduate students. The presentation, by one of the masters of the subject and Abel Prize winner in 2005, is clear and concise, but not dry. The book starts with the basics and quickly moves on to topics such as matrix analysis, convexity, duality, normed linear spaces, and numerical solution of linear systems and eigenvalue problems. Sixteen short appendices discuss interesting special topics and complement in detail the eighteen main chapters, making the book suitable as a reference too.
In all, an informative and useful book, distinguished by its blend of theory and applications, which fulfills its goals admirably.
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.
Preface to the First Edition.
3. Linear Mappings.
5. Determinant and Trace.
6. Spectral Theory.
7. Euclidean Structure.
8. Spectral Theory of Self-Adjoint Mappings.
9. Calculus of Vector- and Matrix-Valued Functions.
10. Matrix Inequalities.
11. Kinematics and Dynamics.
13. The Duality Theorem.
14. Normed Linear Spaces.
15. Linear Mappings Between Normed Linear Spaces.
16. Positive Matrices.
17. How to Solve Systems of Linear Equations.
18. How to Calculate the Eigenvalues of Self-Adjoint Matrices.
Appendix 1. Special Determinants.
Appendix 2. The Pfaffian.
Appendix 3. Symplectic Matrices.
Appendix 4. Tensor Product.
Appendix 5. Lattices.
Appendix 6. Fast Matrix Multiplication.
Appendix 7. Gershgorin's Theorem.
Appendix 8. The Multiplicity of Eigenvalues.
Appendix 9. The Fast Fourier Transform.
Appendix 10. The Spectral Radius.
Appendix 11. The Lorentz Group.
Appendix 12. Compactness of the Unit Ball.
Appendix 13. A Characterization of Commutators.
Appendix 14. Liapunov's Theorem.
Appendix 15. The Jordan Canonical Form.
Appendix 16. Numerical Range.