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MATH 4370 - 7370

MATH 4370-7370
Linear Algebra and Matrix Analysis
Tentative schedule

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.

  1. Matrices are everywhere.
  2. Matrices are everywhere, continued. Brief review of linear algebra (Appendix A in the notes).
  3. Brief review of linear algebra, continued. A bestiary of matrices. Block matrices.
  4. Eigenvalues, eigenvectors, eigenpairs. Characteristic polynomial. Cayley-Hamilton theorem? Algebraic multiplicity.
  5. Similarity.
  6. Left and right eigenvectors. Geometric multiplicity. Diagonalization.
  7. Gershgorin’s disks theorem.
  8. More on Gershgorin.
  9. Unitary matrices.
  10. Unitary matrices, continued. QR decomposition.
  11. Schur decomposition.
  12. Schur decomposition, continued. Spectral theorem for normal matrices.
  13. The Jordan canonical form.
  14. Singular value decomposition.
  15. Singular value decomposition, continued.
  16. Pseudoinverse.
  17. Vector norms and matrix norms.
  18. Matrix norms, continued. Condition number.
  19. Matrix norms and the SVD.
  20. Nonnegative matrices.
  21. Nonnegative matrices, continued. Perron-Frobenius theorem.
  22. Perron-Frobenius theorem, continued.
  23. Primitive matrices.
  24. Primitive matrices, continued. Stochastic matrices. Applications.
  25. M-matrices.