2- Course Aim :-

The Topics of this course include:- Characteristic and minimal polynomial. Eigenvalues, field of values. 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. Singular value decomposition, generalized inverse, projections, and applications. Positive matrices, Perron-Frobenius theorem. Markov chains and stochastic matrices. M-matrices. Structured matrices (Toeplitz, Hankel, Hessenberg). Matrices and optimization (e.g., linear complementarity problem, conjugate gradient). 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.