The effect of perturbation of an off-diagonal entry pair on the geometric multiplicity of an eigenvalue

The change in geometric multiplicity, associated with perturbing a particular pair of off-diagonal entries, in a combinatorially symmetric matrix is investigated. First, we focus upon a Hermitian matrix, with a general graph, when the perturbation is Hermitian. This generalizes prior work in case the graph is a tree; the possibilities are much richer. Then, we turn to general matrices over a field. In both cases, by classifying the incident vertices, all possibilities for the change in geometric multiplicity are identified. Then, examples are given to show that each possibility may actually occur. In some instance, the change in geometric multiplicity depends upon the numerical value of the perturbation, and it matters that both off-diagonal entries change. However, in most instances, the change is qualitatively determined.