Checking date: 10/09/2020

Course: 2020/2021

Advanced methods in matrix analysis
(15445)
Study: Master in Mathematical Engineering (70)
EPI

Coordinating teacher: MORO CARREÑO, JULIO

Department assigned to the subject: Department of Mathematics

Type: Compulsory
ECTS Credits: 6.0 ECTS

Course:
Semester:

Students are expected to have completed
Linear Algebra Calculus
Competences and skills that will be acquired and learning results.
1- Knowledge, applications, and use of basic matrix factorizations: LU, Cholesky, and QR. 2- Knowledge, applications, and use of canonical forms under similarity: Schur and Jordan forms. 3- Spectral Theory, applications, and use of normal matrices and its particular cases: Hermitian/antihermitian matrices, orthogonal and unitary matrices. 4- The use of Min-max theorem for Hermitian matrices and consequences. 5- Knowledge, applications, and use of the Singular Value Decomposition, in particular in approximation problems. 6- Knowledge, applications, and use of Moore-Penrose pseudoinverses in the solution of least squares problems. 7- Knowledge, applications, and use of matrix norms. 8- Knowledge, applications, and use of basic matrix perturbation results.
Description of contents: programme
1- Basics on Matrix Analysis. 1.1- Block partitioned matrices and block operations 1.2- Schur complement and properties. 1.3- Projection matrices and orthogonal projection matrices. 1.4- Basics on eigenvalues and eigenvectors. Diagonalizable matrices. 2- Vector and matrix norms. 2.1- Vector norms. Monotone and absolute norms. 2.2- Important vector norms: 1, 2, and infinity. 2.3- Consistent matrix norms. Important examples: 1, 2, infinity, and Frobenius. 2.4- Convergent matrices and spectral radius. 3- LU and QR factorizations. 3.1- LU factorization and Schur complement. 3.2- Unitary matrices. 3.3- QR factorization and the Gram-Schmidt method. 4- Canonical forms under similarity. 4.1- Block diagonalization and matrix Sylvester equations. 4.2- Schur Form. 4.3- Jordan canonical form. 5- Normal Matrices. Hermitian Matrices. 5.1- Spectral theorem of normal matrices. 5.2- Variational characterizations of eigenvalues of Hermitian matrices: the min-max theorem. 5.3- Eigenvalue interlacing in Hermitian matrices. 5.4- Sylvester's inertia theorem for Hermitian matrices. 6- Singular value decomposition and pseudoinverses. 6.1- The theorem of the singular value decomposition. 6.2- Optimal approximation by matrices with smaller rank. 6.3- Pseudoinverses. Moore-Penrose pseudoinverse. 6.4- Applications to least squares problems. 7- Matrix perturbation theory. 7.1- Perturbation of solutions of linear systems: the condition number of a matrix. 7.2- Perturbation of eigenvalues and invariant subspaces: general theory 7.3- Perturbation of eigenvalues and invariant subspaces: the symmetric case. 7.4- Perturbation of singular values and vectors
Learning activities and methodology
1- Theory classes delivered by the course instructor: the key theoretical results will be presented. 2- Problem solving by the students on their own. 3- Individual meetings of the instructor with students seeking guidance on the contents of the course.
Assessment System
• % end-of-term-examination 60
• % of continuous assessment (assigments, laboratory, practicals...) 40
Basic Bibliography
• G. W. Stewart and J-G. Sun. Matrix Perturbation Theory. Academic Press. 1991
• R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press. 1985
• R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press. 1991