  We study the real algebraic variety of real symmetric matrices with
eigenvalue multiplicities determined by a partition. We present formulas for
the dimension and Euclidean distance degree. We give a parametrization by
rational functions. For small matrices, we provide equations; for larger
matrices, we explain how to use representation theory to find equations. We
describe the ring of invariants under the action of the orthogonal group. For
the subvariety of diagonal matrices, we give the degree.
