# Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with degeneracies into the 1000s. Since any unitary homomorphism on the eigenspace associated with an eigenvalue is a valid solution, I am concerned about how to select the eigenvectors. I would prefer if the basis of each eigenspace had compact representations in the basis that $A$ is given in. To this end, I am looking for simple preconditioning rules that I could use to bias the procedure towards compact eigenvectors.