Tridiagonal realization of the anti-symmetric Gaussian $\beta$-ensemble

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(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(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 Pruumlfer phases of the random matrices. (C) 2010 American Institute of Physics. [doi:10.1063/1.3486071]