Universal behavior of large random matrices

We have already mentioned that when random matrices become large, they begin to display some interesting behaviors. However, what do we mean by this and why is it useful to us as data scientists?

The interesting behavior that we see is that the statistical properties of their eigen-decompositions or singular-value decompositions become universal. By universal, we mean that the same behavior is seen across many different matrices. In the case that we’re going to illustrate, it means that the statistical characteristics of the eigen-decomposition of any large square random matrix are the same.

It is worth recalling from Chapter 3 that eigen-decompositions of square matrices are an important and flexible way of representing any square matrix. So, universality in parts of the eigen-decomposition of large random matrices also means that calculations and algorithms involving the matrices will have universal aspects to them. It won...