Advanced Linear Algebra

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Every Fall

An advanced course in Linear Algebra and applications.


Undergraduate linear algebra at the level of MATH 4305, and ability to write rigorous proofs at the level of MATH 2106.

Course Text: 

Material will be selected from the following books, and supplemented by classnotes as needed. Books: "Matrix Analysis" and "Topics in Matrix Analysis" by Horn & Johnson, "Advanced Linear Algebra" by Roman,  "Linear Algebra and its applications" by Lax.

Topic Outline: 

[Items 1)-5) to be covered every time. These may take between 50% and 75% of the time. The remaining portion of time should be spent on selected topics according to the instructor's interests; see 6) below.]

  1. Characteristic and minimal polynomial. Eigenvalues, field of values.
  2. Similarity transformations: Diagonalization and Jordan forms over arbitrary fields. Schur form and spectral theorem for normal matrices. Quadratic forms and Hermitian matrices: variational characterization of the eigenvalues, inertia theorems.
  3. Singular value decomposition, generalized inverse, projections, and applications.
  4. Positive matrices, Perron-Frobenius theorem. Markov chains and stochastic matrices. M-matrices.
  5. Structured matrices (Toeplitz, Hankel, Hessenberg). Matrices and optimization (e.g., linear complementarity problem, conjugate gradient).
  6. Other topics and applications depending on the interest of the instructor. Examples are Krylov subspaces, tensor and multilinear algebra, integer matrices, Schur complement, matrix equations and inequalities, polar factorization and proper orthogonal decomposition, search algorithms, applications to signal and image processing, matrices depending on parameters, eigenvalues and singular value inequalities, functions of matrices, etc.