Linear Algebra and Matrix Analysis
This is a rough schedule for the course, indicative of the order and content of the lectures. Rather than refer to the exact date at which a course is given, I indicate the lecture number, knowing that the course consists of 25 lectures most years.
- Matrices are everywhere.
- Matrices are everywhere, continued. Brief review of linear algebra (Appendix A in the notes).
- Brief review of linear algebra, continued. A bestiary of matrices. Block matrices.
- Eigenvalues, eigenvectors, eigenpairs. Characteristic polynomial. Cayley-Hamilton theorem? Algebraic multiplicity.
- Left and right eigenvectors. Geometric multiplicity. Diagonalization.
- Gershgorin’s disks theorem.
- More on Gershgorin.
- Unitary matrices.
- Unitary matrices, continued. QR decomposition.
- Schur decomposition.
- Schur decomposition, continued. Spectral theorem for normal matrices.
- The Jordan canonical form.
- Singular value decomposition.
- Singular value decomposition, continued.
- Vector norms and matrix norms.
- Matrix norms, continued. Condition number.
- Matrix norms and the SVD.
- Nonnegative matrices.
- Nonnegative matrices, continued. Perron-Frobenius theorem.
- Perron-Frobenius theorem, continued.
- Primitive matrices.
- Primitive matrices, continued. Stochastic matrices. Applications.