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

We derive efficient recursive formulas giving the exact distribution of the largest eigen- value for finite dimensions 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 show that the Tracy-Widom laws can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications. Thus, the largest eigenvalue distribution for Wishart and Gaussian matrices can be approximated simply by a shifted Gamma distribution with known parameters.