Matristeori

**Valid from:** Autumn 2019**Decided by:** FN1/Anders Gustafsson**Date of establishment:** 2014-04-22

**Division:** Mathematics**Course type:** Third-cycle course**Teaching language:** English

The aim of the course is to give knowledge about the structure of finite dimensional linear maps, and their representing matrices, which is necessary for research in mathematics and, e.g., control theory.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to understand and independently explain the theory of matrix functions, in particular polynomials, and its connection to the Jordan normal form.
- be able to describe different types of vector and matrix norms, and to compute or estimate them with as well as without computer support.
- be familiar with the common classes of normal matrices and their properties.

*Competences and Skills*

For a passing grade the doctoral student must

- independently be able to characterize and use different types of matrix factorizations.
- with access to literature be able to integrate methods and approaches from the different parts of the course in order to solve problems and answer questions within the framework of the course.
- be able to judge which numerical solution method to a given problem best fulfils requirements of speed and exactness.
- with access to literature be able to write Matlab programs for the solution of mathematical problems within the course.
- orally and in writing, with clear logic and with proper terminology be able to explain the solution to a mathematical problem within the course.
- with access to the resources of a library be able independently to assimilate and sum up the contents of a text in technology in which matrix theoretical methods are used.

Matrices and determinants. Linear spaces. Spectral theory.The Jordan normal form. Matrix factorizations. Matrix polynomials and matrix functions. Norms. Scalar products. Singular values. Normal matrices. Quadratic and Hermitian forms. The Least Squares method and pseudo inverses. Direct products of Matrices. Matrices with polynomial elements. Non-negative matrices.

- math-aho & math-vuf: Matrix Theory. KFS, 2013.
- math-aho & math-vuf: Matrix Theory. Studentlitteratur, 2014.

The book covers the part of the course which coincides with the second cycle course. For postgraduate students additional material will be provided. Both editions work.

**Types of instruction:** Lectures, exercises

**Examination formats:** Written exam, oral exam, written assignments.
The assignments are two minor programming exercises in Matlab and Maple, respectively.**Grading scale:** Failed, pass**Examiner:**

The tertiary course coincides with the secondary course FMA120/FMAN70 Matrix theory except that the tertiary course also encompasses three subjects which the doctoral student may choose from five alternatives. Replaces FMA120F.

**Course coordinators:**