Tridiagonal realization of the antisymmetric Gaussian {beta}-ensemble
- Department of Mathematics, University of Washington, Seattle, Washington 98195 (United States)
The Householder reduction of a member of the antisymmetric Gaussian unitary ensemble gives an antisymmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter {beta}, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of (q{sub i}), the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the antisymmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real antisymmetric tridiagonal matrices, its eigenvalues, and (q{sub i}). The third proof maps matrices from the antisymmetric Gaussian {beta}-ensemble to those realizing particular examples of the Laguerre {beta}-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pruefer phases of the random matrices.
- OSTI ID:
- 21476492
- Journal Information:
- Journal of Mathematical Physics, Vol. 51, Issue 9; Other Information: DOI: 10.1063/1.3486071; (c) 2010 American Institute of Physics; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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