Eigenvalues and Eigenvectors of Tau Matrices with Applications to Markov Processes and Economics

Abstract

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called ${\tau }_{ϵ,\varphi }$ algebra, a generalization of the more known $\tau$ algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator $T_{n,ϵ,\varphi }$ of the ${\tau }_{ϵ,\varphi }$ algebra. In particular, we derive the asymptotics for the outliers of $T_{n,ϵ,\varphi }$ and the associated eigenvectors; we obtain equations for the eigenvalues of $T_{n,ϵ,\varphi }$, which provide also the eigenvectors of $T_{n,ϵ,\varphi }$; and we compute the full eigendecomposition of $T_{n,ϵ,\varphi }$ in the specific case $ϵ\varphi =1$. 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.