Johnson C.R., Saiago C.M. Eigenvalues, Multiplicities and Graphs

Cambridge: Cambridge University Press, 2018. — 315 p.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.