Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.
LEARNING OUTCOMES
A student who successfully completes the course will be able to: be familiar with advanced concepts about matrices over field, understand basic results about matrices over commutative rings, apply computational techniques for generalized eigenvalue problems and for computing general functions of a matrix.
Credits: 5
Schedule: 26.10.2020 - 11.12.2020
Teacher in charge (valid 01.08.2020-31.07.2022): Vanni Noferini
Teacher in charge (applies in this implementation): Vanni Noferini
Contact information for the course (valid 02.10.2020-21.12.2112):
Main Lecturer: Prof Vanni Noferini (name.surname@aalto.fi)
Teaching Assistant: Mr Lauri S. Nyman (name.s.surname@aalto.fi)
CEFR level (applies in this implementation):
Language of instruction and studies (valid 01.08.2020-31.07.2022):
Teaching language: English
Languages of study attainment: English
CONTENT, ASSESSMENT AND WORKLOAD
Content
Valid 01.08.2020-31.07.2022:
Matrices over rings, Smith theorem, polynomial matrices, linearizations, canonical forms, theory and computation of functions of matrices, analytic matrix-valued functions, non-negative matrices.
Applies in this implementation:
The course will provide an overview of several topics within matrix theory, some of a more algebraic nature and some of a more computational flavour. Students of both pure and applied mathematics (or strictly related subjects) can benefit from the course. Knowledge of basic linear algebra (e.g. knowing how to solve linear systems, or what the spectral theorem says) is assumed. Knowledge of basic abstract algebra (e.g. knowing what a zero divisor is, or being familiar with the notion of ideal) is also expected, although we will anyway review (or give if not already known) many of the necessary definitions.
Tentatively, we will cover the following topics:- Matrices over commutative rings: basic results
- The Hermite normal form for matrices over a Bezout domain
- The Smith normal form for matrices over a principal ideal domain
- The Jordan canonical form for square matrices over an algebraically closed field
- Pencils over a field: an overview
- The Rellich decomposition of Hermitian analytic matrices
- Polynomial matrices: eigenvalues and minimal bases
- Matrix functions: theory and computation
- The Perron-Frobenius Theorem for non-negative square matrices
To some extent, the actual syllabus can be adapted to the students' background and feedback.
- Matrices over commutative rings: basic results
Assessment Methods and Criteria
Valid 01.08.2020-31.07.2022:
Either written exam or project+presentation (student's choice)
Applies in this implementation:
Option A (warmly recommended): 2 homework sets weighted 20% of the final mark each + 1 final project weighted 60% of the final mark. Possible topics for the final project will be suggested during the course.
Option Z: 100% of the mark based on one devilish, depressingly difficult, maddening final exam
Workload
Valid 01.08.2020-31.07.2022:
24 hours lectures, 6 hours exercises, 100 hours individual study
DETAILS
Study Material
Valid 01.08.2020-31.07.2022:
Primarily lecture notes. Suggested further materials such as books will be provided during the lectures.
Prerequisites
Valid 01.08.2020-31.07.2022:
MS-C134x Linear Algebra, MS-C1081 Abstract Algebra
Registration for Courses
Valid 01.08.2020-31.07.2022:
Via WebOodi.
Applies in this implementation:
In 2020, the course is fully online.
- Teacher: Noferini Vanni
- Teacher: Nyman Lauri