What's the distribution of eigenvalues of a real and symmetric Toeplitz matrix?

Question

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

Answer 1

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.