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

英文摘要：

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