Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution

Understand

We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE).

• In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy-Widom law can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.