Numerical methods for large scale matrices: sparse matrices, structured matrices. Graph theory and permutation techniques for sparse matrices analysis.

Matrix inverse by low rank corrections. Matrix inverse by block partitioning. Schur complements and Woodbury-Sherman-Morrison formula.

Kronecker product. Sylvester matrix equation and Kronecker sum.

QR factorization of sparse matrices.

Integral equations, discretization and convolution. Fast Fourier Transform (FFT) and its applications in matrix algebra. Toeplitz matrices, generating function, spectrum, equidistribution and Szegö-Tyrtysnikov theorem.

Convergence theory for stationary iterative methods for linear systems. Perron-Frobenius theory for nonnegative matrices. Regular splitting. Spectral radius and localization of eigenvalues. 

Iterative methods of minimization for the solution of linear systems. Non-stationary methods. Methods with optimum step length. Method of steepest descent. Conjugate gradient. Convergence analysis via matrix spectrum analysis. 

Preconditioning techniques. 

Matlab laboratory exercises.