Exact Largest Eigenvalue Distribution for Doubly Singular Beta Ensemble

TL;DR

This paper derives an exact formula for the distribution of the largest eigenvalue in the doubly singular beta ensemble, extending previous work on matrix variate distributions and Wishart matrices.

Contribution

It provides a simple, exact expression for the largest eigenvalue distribution in the doubly singular beta ensemble with identity scale matrix, linking it to Roy's statistic.

Findings

Exact distribution formula derived for the largest eigenvalue.

Distribution expressed via Wishart matrices and Roy's statistic.

Simplifies computation of eigenvalue distributions in singular beta ensembles.

Abstract

In \cite{Diaz} beta type I and II doubly singular distributions were introduced and their densities and the joint densities of nonzero eigenvalues were derived. In such matrix variate distributions $p$, the dimension of two singular Wishart distributions defining beta distribution is larger than $m$ and $q$, degrees of freedom of Wishart matrices. We found simple formula to compute exact largest eigenvalue distribution for doubly singular beta ensemble in case of identity scale matrix, $\Sigma=I$. Distribution is presented in terms of existing expression for CDF of Roy's statistic: $\lambda_{max} \sim max \ eig\left\{ W_q(m, I)W_q(p-m+q, I)^{-1}\right\}$, where $W_k(n, I)$ is Wishart distribution with $k$ dimensions, $n$ degrees of freedom and identity scale matrix.