This handout provides the essential elements needed to understand finite random matrix theory. A cursory observation should reveal that the tools for infinite random matrix theory are quite different from the tools for finite random matrix theory. Nonetheless
1 Largest Eigenvalue Distributions In this section, the distributions of the largest eigenvalue of matrices in the β-ensembles are studied. Histograms are created first by simulation, then by solving the Painlev´e II nonlinear differential equation
