Eigenvalue Multiplicities in Principal Submatrices

Let λ be an eigenvalue of an n-by-n matrix A. The allowable patterns of geometric multiplicities of λ as an eigenvalue of A and its principal submatrices is explored. A graphical hierarchy for succinctly reporting the possible patterns is defined. Special attention is paid to the case in which A is Hermitian. Classical interlacing already imposes much structure on the hierarchies in the Hermitian case. Here, all the known constraints, some old and some new, on the geometric multiplicity hierarchies of Hermitian matrices are listed. Some differences between allowed hierarchies for real symmetric matrices and Hermitian matrices are also discussed.