Eigenvalues and Eigenvectors of Tau Matrices with Applications to Markov Processes and Economics
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In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called τ_ε,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,ε,φ of the τ_ε,φ algebra. In particular, we derive the asymptotics for the outliers of T_n,ε,φ and the associated eigenvectors; we obtain equations for the eigenvalues of T_n,ε,φ, which provide also the eigenvectors of T_n,ε,φ; and we compute the full eigendecomposition of T_n,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.
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