Smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles
Abstract
We compute the exact and limiting smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles not covered by previous studies. In the general $\beta$ case, these distributions are given by multivariate hypergeometric ${}_2F_{1}^{2/\beta}$ functions, whose behavior can be analyzed asymptotically for special values of $\beta$ which include $\beta \in 2\mathbb{N}_{+}$ as well as for $\beta = 1$. Interest in these objects stems from their connections (in the $\beta = 1,2$ cases) to principal submatrices of Haar-distributed (orthogonal, unitary) matrices appearing in randomized, communication-optimal, fast, and stable algorithms for eigenvalue computations \cite{DDH07}, \cite{BDD10}.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2010
- DOI:
- 10.48550/arXiv.1009.4677
- arXiv:
- arXiv:1009.4677
- Bibcode:
- 2010arXiv1009.4677D
- Keywords:
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- Mathematics - Probability;
- Computer Science - Distributed;
- Parallel;
- and Cluster Computing;
- Computer Science - Numerical Analysis;
- Mathematical Physics;
- 60B20;
- 15B52
- E-Print:
- 15 pages, 6 figures