Among the n eigenvalues of an n-by-n matrix may be several repetitions (the number of which counts toward the total of n). For general matrices over a general field, these multiplicities may be algebraic (the number of appearances as a root of the characteristic polynomial) or geometric (the dimension of the corresponding eigenspace). These multiplicities are quite important in the analysis of matrix structure because of numerical calculation, a variety of applications, and for theoretical interest. We are primarily concerned with geometric multiplicities and, in particular but not exclusively, with real symmetric or complex Hermitian matrices, for which the two notions of multiplicity coincide.

It has been known for some time, and is not surprising, that the arrangement of nonzero entries of a matrix, conveniently described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues. Much less limited by this information are either the algebraic multiplicities or the numerical values of the (distinct) eigenvalues. So, it is natural to study exactly how the graph of a matrix limits the possible geometric eigenvalue multiplicities.

Organized study of “eigenvalues, multiplicities and graphs” really began in the 1990s, though two earlier papers, [P] and [Wie], play an important role, including motivational. There had also been considerable interest in the eigenvalues of particular matrices with a given graph, such as the adjacency or Laplacian matrix. It was recognized early that the theory is most rich in case the graph is minimally connected, i.e., a tree. For this reason, the theory is relatively well developed for trees. However, in recent papers and in the preparation of this monograph, there has been an opportunity to identify more clearly which ideas have analogs for nontrees and for which ideas trees are essential. We have also recently noticed that for trees, and sometimes for general graphs, ideas about real symmetric/complex Hermitian matrices carry over to geometric multiplicities in general matrices over a field, sometimes under a diagonalizability hypothesis. This is an important advance that we have included herein (Chapter 12); the proofs are necessarily very different, and we have also included earlier proofs for the symmetric case, which are of interest for themselves and support other work.