A random matrix decimation procedure relating β= 2/(r+1) to β= 2(r+1)

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Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case r=1 of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as β-ensembles. The inter-relations give that the joint distribution of every (r+1)-st eigenvalue in certain β-ensembles with β= 2/(r+1) is equal to that of another β-ensemble with β= 2(r+1). The proof requires generalizing a conditional probability density function due to Dixon and Anderson.