A Second Course in Linear Algebra

Stephan Ramon Garcia is W. M. Keck Distinguished Service Professor and Professor of Mathematics at Pomona College, California. He is the author of two books and over seventy research articles in operator theory, complex analysis, matrix analysis, number theory, discrete geometry, and other fields. He is on the editorial boards of the Proceedings of the American Mathematical Society, Involve, and the American Mathematical Monthly. He received three National Science Foundation (NSF) research grants as principal investigator, five teaching awards from three different institutions, and was twice nominated by Pomona College for the Council for Advancement and Support of Education (CASE) US Professors of the Year award. Roger A. Horn was Research Professor of Mathematics at the University of Utah. His publications include Matrix Analysis, 2nd edition (Cambridge, 2012) and Topics in Matrix Analysis (with Charles A. Johnson, Cambridge, 1991), as well as more than 100 research articles in matrix analysis, statistics, health services research, complex variables, probability, differential geometry, and analytic number theory. He was Editor of The American Mathematical Monthly (1996–2001), the Mathematics Association of America Spectrum book series (1992–5), and the MAA Carus Mathematical Monographs (2002–5). He has also served on the editorial boards of the SIAM Journal of Matrix Analysis, Linear Algebra and its Applications, and the Electronic Journal of Linear Algebra.

Preliminaries; 1. Vector spaces; 2. Bases and similarity; 3. Block matrices; 4. Inner product spaces; 5. Orthonormal vectors; 6. Unitary matrices; 7. Orthogonal complements and orthogonal projections; 8. Eigenvalues, eigenvectors, and geometric multiplicity; 9. The characteristic polynomial and algebraic multiplicity; 10. Unitary triangularization and block diagonalization; 11. Jordan canonical form; 12. Normal matrices and the spectral theorem; 13. Positive semidefinite matrices; 14. The singular value and polar decompositions; 15. Singular values and the spectral norm; 16. Interlacing and inertia; Appendix A. Complex numbers.