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Anti-symmetric Gaussian unitary matrices have recently been connected to the tiling of the half hexagon by three species of rhombi, as well as to classical complex Lie algebras. Their Householder reduction gives an anti-symmetric tridiagonal matrix with independent elements, which permit $\beta$ generalizations. We computed their eigenvalue PDFs, as well as the distribution of the first components of the eigenvectors, in three ways. The first one involves an inductive construction based on bordering of a family of matrices, the second one--a Jacobian computation for the change of variables for real anti-symmetric tridiagonal matrices, and the last one--a linear algebra transformation to a particular example of $\beta$-Laguerre ensembles. We will sketch the first two and present the third. This is joint work with Peter Forrester.