Matrix theory. Unitary, Hermitian, positive definite, normal matrices.

Normal forms: Schur, Jordan

Spectral characterization of unitary, Hermitian, positive definite, normal matrices via the Schur form

Eigenvalues: localization (Theorems by Gerschgorin I, II, III)

Vector norms, matrix norms, induced norms (relation between spectral radius and induced norms)

Theorem of topological equivalence in finite-dimensional vector spaces

Elementary matrices (spectral analysis, inverse): Gauss, Householder

Numerical solution of linear systems: coefficient matrices in special form (unitary, triangular etc)

Conditioning and the problem and stability of the algorithms

Numerical solution of linear systems: Gaussian elimination, pivoting, QR factorization

Choleski algorithm for positive definite matrices

Shermann-Morrison-Woodbury formula (updating efficient techniques)

Numerical stability of the direct algorithms

Iterative methods: general theory, Jacobi and Gauss-Seidel (methods and convergence analysis)

Evaluation of a polynomial at a point. Interpolation. Vandermonde Matrix