TITLE: What's the distribution of eigenvalues of a real and symmetric Toeplitz matrix?
QUESTION [7 upvotes]: 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

REPLY [4 votes]: 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.