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

TL;DR

This paper investigates the eigenvalues and eigenvectors of tau matrices within the $ au_{ ext{epsilon,phi}}$ algebra, deriving asymptotics and explicit decompositions, with applications to Markov processes, economics, and related stochastic models.

Contribution

It introduces new spectral analysis results for tau matrices, including asymptotics and eigendecomposition, extending previous algebraic frameworks and applying findings to economic and stochastic models.

Findings

Derived eigenvalue asymptotics for tau matrices.

Obtained explicit eigendecomposition in special cases.

Applied results to queuing, random walks, and economic models.

Abstract

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called $\tau_{\varepsilon,\varphi}$ 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 $T_{n,\varepsilon,\varphi}$ of the $\tau_{\varepsilon,\varphi}$ algebra. In particular, we derive the asymptotics for the outliers of $T_{n,\varepsilon,\varphi}$ and the associated eigenvectors; we obtain equations for the eigenvalues of $T_{n,\varepsilon,\varphi}$, which provide also the eigenvectors of $T_{n,\varepsilon,\varphi}$; and we compute the full eigendecomposition of $T_{n,\varepsilon,\varphi}$ in the specific case $\varepsilon\varphi=1$. 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.