@TechReport{	  it:2020-003,
  author	= {Sven-Erik Ekstr{\"o}m and Carlo Garoni and Adam Jozefiak
		  and Jesse Perla},
  title		= {Eigenvalues and Eigenvectors of Tau Matrices with
		  Applications to {M}arkov Processes and Economics},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2020},
  number	= {2020-003},
  month		= aug,
  abstract	= {In the context of matrix displacement decomposition, Bozzo
		  and Di Fiore introduced the so-called
		  $\tau_{\epsilon,\phi}$ 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 $T_{n,\epsilon,\phi}$ of the
		  $\tau_{\epsilon,\phi}$ algebra. In particular, we derive
		  the asymptotics for the outliers of $T_{n,\epsilon,\phi}$
		  and the associated eigenvectors; we obtain equations for
		  the eigenvalues of $T_{n,\epsilon,\phi}$, which provide
		  also the eigenvectors of $T_{n,\epsilon,\phi}$; and we
		  compute the full eigendecomposition of
		  $T_{n,\epsilon,\phi}$ in the specific case
		  $\epsilon\phi=1$. 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.}
}