The Eigenvalues of Random Matrices

Although one can see earlier glimmers (e.g., [29]), the study of random matrices as such originated in statistics in Wishart’s 1928 consideration of random sample covariance matrices [54], in numerical analysis in von Neumann and collaborators’ work in the 1940s on numerical methods for solving linear systems [49, 23], and in nuclear physics in Wigner’s 1955 introduction of random matrices as models for atoms with heavy nucleii [52, 53]. Since then, random matrix theory has found countless applications both within mathematics and in science and engineering. While using random matrices as statistical models in the presence of uncertainty is perhaps the most obvious way to go, there is a more fundamental reason for those who study and use matrices to know something about random matrices: recognizing what’s typical. First mathematical questions tend to be about what is possible: How large or small can the eigenvalues be? How long might the algorithm take? How many edges can a graph have and still contain no triangles? But for some purposes, what is possible is less relevant than what is typical: How large or small do the eigenvalues tend to be? How long does the algorithm usually take? For what size graphs is it fairly common to have no triangles? The single problem in random matrix theory which has received the most attention is that of understanding the eigenvalues of random matrices. Of course, this is not actually a single problem, but a huge class of problems: There are many ways to build a random matrix (leading to many drastically different eigenvalue distributions), and there are many different things to understand about any given random matrix model. In this note, I will give a broad overview of some random matrix models and some of what is known about their eigenvalues.