Characterize correlation matrices with multiple prescribed null vectors

Determine a general necessary and sufficient condition for the existence of an m-dimensional correlation matrix whose null space contains multiple prescribed constraint vectors, beyond the special case in which the constraint vectors have nonzero entries at different indices.

Background

The Gaussian benchmarking prior requires constructing a correlation matrix R_u satisfying R_u W = 0, so that the prior—and consequently the posterior—is supported on the benchmarking constraint set. For a single constraint vector, an exact characterization is available: such a correlation matrix exists if and only if the vector is balanced. When there are multiple constraint vectors, the paper notes that a general characterization had not been available. The authors then identify a special benchmarking setting—constraint vectors with nonzero entries at different indices—in which checking the balance condition for each vector is both necessary and sufficient. The unresolved problem is to obtain the corresponding general criterion without that special structural assumption.