Random matrix theory is very exciting and has tools for problems we encounter again and again across a vast expanse of the math landscape. Already in exploring quadratic convexity and algorithms for deciding it, we see a need for sampling methods (with accompanying estimates for distributions of derived quantities). Here is a great survey paper by a former Cal Poly student (et al), now a professor at Boulder. This is just one area of this newish industry within mathematics (henceforth RMT).
Numerical linear algebra and, more generally, numerical algebraic geometry are going to come in very handy. Here is an introduction to the latter. Here is a great text on matrix analysis with mostly constructive proofs, an excellent handbook. And this book by Dym has useful but not-so-well-known results.
And what a great time to be looking into this topic; here is an updated discussion of open problems in RMT and integrable systems.