Topic outline

  • General

    Welcome to the Matrix Theory course home page. The goal of the course is to discuss matrices at a level more advanced than in basic courses, focusing also on the interactions of matrix theory with abstract algebra on one hand and numerical analysis on the other hand.

    Teachers

    Main lecturer: Vanni Noferini, vanni (dot) noferini (at) aalto (dot) fi
    Teaching assistant: Lauri Nyman, lauri (dot) s (dot) nyman (at) aalto (dot) fi


    Timetable

    Lectures: generally Monday 14:15 to 16 (from 24.10 to 28.11) and Thursday 14:15 to 16 (from 27.10 to 1.12), all in room M2 (M233)

    Exception: in the second week of teaching the lecture will not be on Thursday 3.11 but on Tuesday 1.11 (still 14:15 to 16 in M2)

    Exercises: Friday 12:15 to 14 (from 4.11 to 2.12)

    Project Presentations: To be agreed on an individual basis.

    Tentative syllabus


    • Matrices over general commutative rings.
    • Matrices over principal ideal domains: Hermite and Smith canonical forms.
    • Matrices and pencils over fields. Canonical forms.
    • Matrices over the ring of analytic functions. Rellich decomposition.
    • Polynomial matrices. Eigenvalues and minimal bases.
    • Theory and computation of matrix functions.
    • Non-negative matrices. Perron-Frobenius theory.


    Passing the course

    The grade will be based on two homework exercise sets and a project. The topic of the project can be anything related to the syllabus; suggestion on possible topics will be given during the course.

    The final grade will be based 40% on the homework and 60% on the project.

    Course material

    Lecture notes, slides, links to video proofs will be posted on MyCourses after each week of lectures. The reason to not publish to material beforehand is for a pedagogical choice: we will start some topics with a bottom-up approach that works best if one has not seen the lecture notes/videos before.

    Model solutions to practice exercises will also be published after the relevant exercise session. On the contrary, texts of practice exercises and of homework exercises will be published in advance. More specific rules related to homework (deadlines etc.) will be announced in due course.

    Recommended books

    The course does not follow exactly any one textbook, but it has been in part inspired by some textbooks.
    For those students who are interested in learning more on matrix theory, below is a short list of books that can be consulted.  The superposition of the course and any single one of these books is partial: some (but not all) of the results in the course have been taken from some of those books, and on the other hand each book goes way beyond the course content, in some direction. Next to each reference, I indicate which chapters of the course are, at least in part, treated in the book (note: the main topic of the book might be something else!). The list is by authors alphabetical order and not by amount of superposition with the course -- the book that superposes the most with the course is Friedland's.

    • W. Brown, "Matrices over commutative rings", Marcel Dekker 1993. (Chapter 1)
    • S. Friedland, "Matrices: Algebra, analysis and applications", World Scientific 2016. (Chapters 2, 3, 5, 6).
    • I. Gohberg, P. Lancaster, L. Rodman, "Matrix polynomials", SIAM 2009. (Chapters 2, 4).
    • N. Higham, "Functions of matrices: Theory and computation", SIAM 2008. (Chapter 5)
    • T. Kato, "Perturbation theory of linear operators", Springer-Verlag 1980. (Chapter 4)

    Possible topics for the final project

    Possibilities include: 

    -reading and reporting on a book chapter/section, or a research article, that develops further some topic treated in the course or that is connected to what was treated in the course

    -developing and reporting on software that implements some of the computations or algorithms that are treated in the course or that are connected to what is treated in the course

    Specific examples of possible projects will be suggested later on; you can also contact the lecturer directly for more tailored options if you already which topic is most of interest to you.