Summary: Let denote the square of the largest singular value of an matrix , all of whose entries are independent standard Gaussian variates. Equivalently, is the largest principal component variance of the covariance matrix , or the largest eigenvalue of a -variate Wishart distribution with degrees of freedom and identity covariance. Consider the limit of large and with . When centered by and scaled by , the distribution of approaches the Tracy-Widom law [C.A. Tracy and H. Widom, J. Stat. Phys. 92, No. 5-6, 809-835 (1998; Zbl 0942.60099)] of order 1, which is defined in terms of a Painlevé II differential equation and can be numerically evaluated and tabulated by software.
Simulations show the approximation to be informative for and as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large multivariate distribution theory may be easier to apply in practice than their fixed counterparts.