# 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.

### Registration

Please write a mail to Alexander Zeilmann until October 20th, 2021 if you consider participating.

If there are too many students interested in the seminar, the list of participants will be chosen at random.

The first meeting, discussing the seminar and answering questions will take place in the first or second 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).

### Organization

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.

### Schedule

• Submission slides: one week in advance of your presentation
• Submission summary: one week after your presentation
• Talks: Depending on the number of participants as block seminar or weekly seminar. Also depending on the current regulations and the wishes of the participants, the seminar will be in person or online.

### Topics

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