Course syllabus - Applied Matrix Analysis

Objectives

The course provides broad knowledge of applications of matrices and of the essential tools of matrix analysis in various areas of engineering and natural sciences. The basic concepts and methods of importance for further study is explained with practical examples from finance, economics, statistics, discrete mathematics and related models from energy, environment and resource optimization, systems analysis, automatic control, computer science and information technology. In addition to training in logical and geometric thinking and the modeling and computing with matrices of particular importance for applications, as well as the capacity for independent analysis and solution of mathematical problems and models is trained.

Learning outcomes

After completing the course, students should be able to

- describe the basic properties of non-negative and stochastic matrices, their connection with graphs and Markov chains and applications in economics, resource optimization, information technology, linear and dynamic programming, decision making, game theory and systems analysis.
- calculate the matrix, canonical forms, functions of matrices and solutions of matrix equations and apply them in studies of system stability and in energy engineering.
- describe different types of matrix and vector norms, and calculate or estimate those with and without computer.
- explore and analyse iterative algorithms for calculating eigenvalues and eigenvectors for various types of matrices with and without computer.
- describe the properties of quadratic forms, projections, spectral theory and their use in quadratic optimization and variational principles and applications in statistics, finance and automatic control.
- analyse the matrix computations in geometrical terms of linear spaces, linear transformations and symmetries.
- explain the concepts within content of the course in a way that is appropriate for the recipient's prior knowledge, and describe a handful of applications.
- describe in detail a freely chosen application area of matrix analysis.

Course content

- Non-negative and stochastic matrices
- Matrix factorisations
- Canonical forms
- Matrix polynomials and matrix functions
- Matrix equations and system stability
- Spectral theory, projections, norms of matrices and vectors
- Scalar
- Singular values
- Quadratic forms
- Quadratic optimization and variational principles
- Iterative algorithms for matrices
- Matrix computations in terms of linear transformations and symmetries; applications of matrix analysis in engineering and natural sciences

Tuition

Lectures, and classes with work individually and in group.

Specific requirements

At least totally 120 credits in the engineering, natural sciences, business administration or economics areas including Basic Calculus Continuation Course, 7.5 credits, of which 3.5 credits must be completed at the beginning of the course, and Basic Vector Algebra, 7.5 credits, of which 5 credits must be completed at the beginning of the course, or the equivalent. In addition Swedish course B/Swedish course 3 and English course A/English course 6 are required. For courses given entirely in English exemption is made from the requirement in Swedish course B/Swedish course 3.

Examination

Project (PRO1), 4.5 credits, marks Pass (G) or Pass with distinction (VG)
Exercise (INL1), 3 credits, marks Pass (G) or Pass with distinction (VG)

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Three-grade scale