I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic that extend the results in that book, or maybe something different.
I have an engineering background but I have interest in pure and applied math. My research focus is on control theory, dynamical systems and optimization. I studied real analysis and fundamentals of functional analysis and measure theory. I also took courses in probability theory, Fourier analysis. I have modest knowledge of abstract algebra (structures).
I'm looking for suggestion of topics that will relate the common domains of all these courses or perhaps give new insight.
The presentation will be 10 mins. I'm expected to spend one day studying that topic.
If your audience has basic knowledge in measure theory/probability theory, you could do a presentation on random matrices.
First, you could introduce the subject by talking about the applications that the study of the distribution of the eigenvalues of random matrices it has in the sciences (particularly physics) and engineering (big data & communications technology). There is a lot of literature which covers this, and a simple google search should bring many results.
Then, if your presentation requires some technical results, you could present a proof (or an outline of the proof) of Wigner's semicircle law, which is widely considered as one of the foundational result of the theory of random matrices (for a proof of Wigner's semicircle law, I recommend section 2.1 of An Introduction to Random Matrices by Anderson, Guionnet, and Zeitouni).
The proof in question uses some nontrivial results in matrix analysis which you have most likely seen during your course, such as the fact that for a normal matrix $X$ (i.e., $XX^*=X^*X$, where $X^*$ denotes the conjugate transpose), we have that $$\text{Tr}(X^k(X^*)^l)=\sum_{i=1}^n(\lambda_i(X))^k\overline{(\lambda_i(X))^l},$$ where $(\lambda_i(X):1\leq i\leq n)$ are the eigenvalues of $X$ (this result depends on the fact that any normal matrix admits a decomposition $UDU^*$, where $U$ is unitary and $D$ is diagonal with the eigenvalues in the diagonal, and that for any matrices $A,B$, we have that $\text{Tr}(AB)=\text{Tr}(BA)$). One possible disadvantage is that this result also involves a lot of combinatorics and probability theory, which your classmates are maybe not that familiar with.