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Can you please help me find the distribution of eigenvalues of a Toeplitz matrix $\mathbf{K}$ that is constructed as follows: $\mathbf{K}=\left[ \begin{array}{cccc} 1 & \rho & \ldots & \, \, \rho^{N-1} \\ \rho & 1 & \ldots & \, \,\rho^{N-2}\\ \vdots & \vdots & \ddots & \vdots \\ \rho^{N-1} & \rho^{N-2} & \ldots & 1 \\ \end{array} \right].$ where $0 \leq \rho < 1$.

Thanks a lot in advance,

Farzad

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    Because this question is purely about the eigenvalues of these matrices, it is more suited for posting on math.SE.2011-08-15

1 Answers 1

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For large $N$, the eigenvalues of $K$ are approximately distributed as $2\pi f(\lambda)$ evaluated at frequencies $-\pi + 2\pi j/N$; $f(\lambda)$ is the spectral density associated to the covariance sequence in your Toeplitz matrix. See for instance Hannan, E.J. Time Series Analysis, Chap. 1 towards the end.

You may find more details in Grenander & Szego, Toeplitz forms and their applications, but I do not have that book at hand and cannot say from memory if it will answer your question more precisely.

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    @Farzad The range $\lambda_j = -\pi + 2\pi j /N$ for $0\leq j \leq N$ doesn't seem right. With that range, I end up with N + 1 values, but the matrix $\mathbf{K}$ will have only ${N}$ eigenvalues. I think it should be 0 \leq j < N.2012-03-22