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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.

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    $\begingroup$ I think this fits better on math.se since you're merely looking for a mathematical topic and not inquiring about, for instance, the best methods to teach said topic. $\endgroup$ – Brendan W. Sullivan Jun 5 '14 at 18:22
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    $\begingroup$ I think extending Horn's book with one day study for a 10 minute presentation is too optimistic. You can maybe try the Perron-Frobenius theorem. Edit: I just realised Horn mentions it in his book. $\endgroup$ – Git Gud Jun 5 '14 at 22:08
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    $\begingroup$ Get Topics in Matrix Analysis by the same authors and pick any topic you like. But I agree with @GitGud, this is not a one day study. $\endgroup$ – Artem Jun 5 '14 at 22:25
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10 minutes is not a lot, especially when looking for an advanced topic. I would not expect to be able to go into much detail with your presentation.

What's from your point of view (as a researcher), the most important theorem regarding and/or using matrices in each of

a) control theory

b) dynamical systems

c) optimization?

Among these, is there one theorem where you would consider the degree of linear algebra to be "advanced"?

To be more precise for a): What's your opinion towards topics like the pole-placement theorem ("Given a polynomial $p$ and some matrices $A, B$, when can you find a matrix $F$ such that $A + BF$ has $p$ as characteristic polynomial?"), transfer matrices/functions, solutions of Riccati equations? I feel that something like chapter 10 of this book may fit to your personal field of interest, but will probably be too unwieldy for such a talk.

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