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I've been searching for a subfield of research to get into and wonder whether random matrix theory could suit me well; it seems like it does, because the stuff I read, and the seminars that I watch online seem to show mathematicians mostly ... doing linear algebra, which I am pretty strong in.

Am I wildly off the mark with the idea that random matrix theory is a lot like doing linear algebra, with random entries in the matrices?

Does random matrix theory have any real-world applications, say, to quantitative finance?

I'm still just beginning to learn about this subfield, so any advice would be greatly appreciated.

I think I am moving away from numerical / analytical partial differential equations, as I don't feel I have strong enough physical reasoning / physics background to do good research in PDEs.

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    $\begingroup$ Random matrices are mathematically interesting. The eigenvalue distribution, for instance, is a standard problem. You might give yourself that problem to solve for $P(\lambda)$ for a random $2 \times 2$ matrix $M$ with the matrix elements $M_{i,j} \sim N(0,1)$. Practically speaking, an additional constraint makes the problem much more interesting, like positive definiteness. For instance, a common practical problem involves making small random perturbations to covariance matrices (guaranteed to be positive definite) for applications in modeling multivariate normal distributions. $\endgroup$ – Peter Leopold Jul 23 '19 at 12:35
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    $\begingroup$ I think linear programming (LP) and semi-definite programming (SDP) are both very fun areas of rich linear algebra with broad application domains. And yes, it has been a while since I've seen a PDE arise anywhere but in a narrow area of physics. (I'd be delighted to be contradicted on this point!) $\endgroup$ – Peter Leopold Jul 23 '19 at 12:40
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Go ahead! Within statistics, random matrices is used in multivariate analysis, a good "intro" is https://www.amazon.com/Aspects-Multivariate-Statistical-Theory-Muirhead/dp/0471769851 but is useful also in many other places. See this list of papers.

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