  We consider the following problem: When do alternate eigenvalues taken from a
matrix ensemble themselves form a matrix ensemble? More precisely, we classify
all weight functions for which alternate eigenvalues from the corresponding
orthogonal ensemble form a symplectic ensemble, and similarly classify those
weights for which alternate eigenvalues from a union of two orthogonal
ensembles forms a unitary ensemble. Also considered are the $k$-point
distributions for the decimated orthogonal ensembles.
