Topics:
1. Eigenvalues and eigenvectors of matrices, invariant subspaces, characteristic polynomial, diagonalizable matrices, minimal polynomial, Cayley-Hamilton Theorem, localization of eigenvalues, Gerschgorin's Theorem.
2. Unitary, Hermitian and skew-Hermitian matrices. Cayley transform. Normal matrices and the Spectral Theorem. The field of values and the numerical radius of a matrix. Bendixson's Theorem.
3. The Singular Value Decomposition. Matrix norms: spectral norm and Frobenius norm. Solution to matrix nearness problems. Applications to signal processing and information retrieval.
4. Moore-Penrose pseudoinverse. Applications to the solution of under- and over-determined systems of linear equations. Other generalized inverses. Applications to data fitting (least-squares approximation).
5. Jordan canonical form. An algorithmic proof. Powers of matrices. Matrix functions. Applications to systems of differential equations.
6. Bilinear and quadratic forms. Hermitian forms. Congruence. Sylvester's Law of Inertia. Rayleigh's principle. Courant-Fischer Theorem. Positive definite and semidefinite matrices. Applications to statistics (covariance matrices) and numerical analysis (PDEs).
7. Nonnegative matrices. The spectral radius. Positive matrices. Directed graphs. Nonnegative irreducible matrices. Perron-Frobenius Theorem. M-matrices. Applications to probability theory (Markov chains), economics (Leontiev's input-output model), and information retrieval (Google's PageRank algorithm).
Note: Depending on the Instructor, section 8 below may be covered instead of 7.
8. Structured matrices: circulant, Toeplitz, Hankel, Cauchy, Vandermonde, others. Block generalizations. Applications in signal processing, image processing, and numerical analysis (PDEs, interpolation).
Textbook:
R. A. Horn and C. R. Johnson: Matrix Analysis, Cambridge University Press (1985; 1991).
Additional readings:
C. D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000.
R. A. Horn and C. R. Johnson, "Topics in Matrix Analysis", Cambridge University Press (1991; 1994).
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences", Academic Press (1979); reprinted by SIAM, 1994.
F. R. Gantmacher, "The Theory of Matrices", vols. I-II, Chelsea (1959; 1971).
D. Serre, "Matrices. Theory and Applications", Springer, 2002.