Derivation of eigenvalues and eigenvectors of the Hamiltonian in the fundamental equation describing online user dynamics

Abstract

The Hamiltonian in the fundamental equations used to model online user dynamics consists of the sum of two matrices that generally cannot be diagonalized simultaneously. Therefore, except for exceptional cases where networks are regular graphs, the eigenvalues of the Hamiltonian cannot be derived from the individual eigenvalues of those two matrices. This paper provides a derivation of the eigenvalues and eigenvectors of the Hamiltonian by considering that of the matrix obtained by squaring the Hamiltonian.