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We present an explicit solution of the eigen-spectrum Toeplitz matrix$C_{ij}= e^{- \kappa |i-j|}$ with $0\leq i,j \leq N$ and apply it to findanalytically the plasma modes of a layered assembly of 2-dimensional electrongas. The solution is found by elementary means that bypass the Wiener-Hopftechnique usually used for this class of problems. It rests on the observationthat the inverse of $C_{ij}$ is effectively a nearest neighbor hopping modelwith a specific onsite energies which can in turn be diagonalized easily.Extensions to a combination of a Toeplitz and Hankel matrix, and to ageneralization of $C_{ij}$, are discussed at the end of the paper.

The Toeplitz matrix $e^{- \kappa |i-j|}$ and its application to a layered electron gas