The combinatorial structure of generalized eigenspaces – from nonnegative matrices to general matrices

Abstract

The Perron–Frobenius spectral theory of nonnegative matrices motivated an intensive study of the relationship between graph theoretic properties and spectral properties of matrices. While for about seventy years research focused on nonnegative matrices, in the past fifteen years the study has been extended to general matrices over an arbitrary field. One of the major original problems in this context is determining the relations between the matrix analytic height characteristic of a matrix and the graph theoretic level characteristic. In this article the history of this problem is reviewed, from its introduction for nonnegative matrices, through its complete solution for nonnegative matrices, to the solution of the generalized version of the problem for general matrices.