Proseminar: Functions of Matrices

Probably the most well-known matrix function (or function of matrix) is the matrix exponential. It is defined by inserting a matrix in the power series of the exponential function $$e^A = \sum_{k=0}^\infty{1 \over k!}A^k.$$ This raises the question: Can we define a matrix sine function by inserting a matrix in the power series of the scalar sine function? The answer is yes: $$ \sin(A) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}. $$ In this seminar, we will discuss when and how we can extend scalar functions to matrices, the properties of these functions and various applications.

There are still topics available! Please write a mail to Alexander Zeilmann, if you are interested.

The first meeting, discussing the seminar and answering questions will take place in the first week of the semester.

Afterward, you write a mail with a ranking of your favorite 2 (or more) papers, that the papers can be assigned in case of collisions (otherwise chosen by lot).

The seminar is targeted mainly at Bachelor students in mathematics and related areas.

Every Student is supposed to introduce one of the listed papers by a 30-minute talk (plus discussion). Additionally, write a small summary with 3 to 5 pages at most (references excluded). You may choose between English or German for your summary, slides and talk.

• Submission slides: one week in advance of your presentation
• Submission summary: one week after your presentation
• Talks: weekly virtual seminar

A few chapters from the book Functions of Matrices by Nicholas Higham.