Eigenvalues and eigenvectors of tau matrices with applications to Markov processes and economics
Sven‐Erik Ekström, Carlo Garoni, Adam Jozefiak, Jesse Perla · Linear Algebra and its Applications · 2021
In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called τε,φ algebra, a generalization of the well known τ algebra. We study the properties of eigenvalues and eigenvectors of the generator Tn,ε,φ of the τε,φ algebra. In particular, we derive the asymptotics for the outliers of Tn,ε,φ and the associated eigenvectors; we obtain equations for the eigenvalues of Tn,ε,φ, which provide also the eigenvectors of Tn,ε,φ; and we compute the full eigendecomposition of Tn,ε,φ in the specific case εφ=1. We also present applications of our results in the context of queuing models, random walks, and diffusion processes, with a special attention to their implications in the study of wealth/income inequality and portfolio dynamics.