PRINCIPAL SUBMATRICES, GEOMETRIC MULTIPLICITIES, AND STRUCTURED EIGENVECTORS

It is a straightforward matrix calculation that if lambda is an eigenv alue of A, x an associated eigenvector and alpha the set of positions in which x has nonzero entries, then lambda is also an eigenvalue of t he submatrix of A that lies in the rows and columns indexed by alpha. A converse is presented that is the most general possible in terms of the data we use. Several corollaries are obtained by applying the main result to normal and Hermitian matrices. These corollaries lead to re sults concerning the case of equality in the interlacing inequalities for Hermitian matrices, and to the problem of the relationship among e igenvalue multiplicities in various principal submatrices.