Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by $$ A_\alpha := \begin{bmatrix} 1 & \alpha & \alpha^2 & \ldots & \ldots &\alpha^{n-1} \\\ \alpha & 1 & \alpha & \ddots & \ddots & \vdots \\\ \alpha^2 & \alpha & \ddots & \ddots & \ddots& \vdots \\\ \vdots & \ddots & \ddots & \ddots & \alpha & \alpha^2 \\\ \vdots & \ddots & \ddots & \alpha & 1 & \alpha \\\ \alpha^{n-1} & \ldots & \ldots & \alpha^2 & \alpha & 1 \end{bmatrix}. $$

We can decompose $A_{\alpha}=U_{\alpha}D_{\alpha}U_{\alpha}^{*}$ where $U_{\alpha}$ is a unitary matrix and $D_{\alpha}$ is a diagonal matrix.

Are the matrices $U_{\alpha}$ and $D_{\alpha}$ explicitly known as a function of $\alpha$ and $n$? Can anyone point me to the right reference.

Thanks!

Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by $$ A_\alpha := \begin{bmatrix} 1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\ \alpha & 1 & \alpha & \ddots & \vdots \\\ \alpha^2 & \alpha & \ddots & \ddots & \alpha^2 \\\ \vdots & \ddots & \ddots & 1 & \alpha \\\ \alpha^{n-1} & \ldots & \alpha^2 & \alpha & 1 \end{bmatrix}. $$

We can decompose $A_{\alpha}=U_{\alpha}D_{\alpha}U_{\alpha}^{*}$ where $U_{\alpha}$ is a unitary matrix and $D_{\alpha}$ is a diagonal matrix.

• Are the matrices $U_{\alpha}$ and $D_{\alpha}$ explicitly known as a function of $\alpha$ and $n$? Can anyone point me to the right reference.

• What is known about the spectral limit distribution of these matrices as $n\to\infty$? More specifically, can we compute the limit moments $$ \gamma_{k}:=\lim_{n\to\infty}{\frac{1}{n}\mathrm{Tr}\Big(A_{\alpha}^{k}\Big)} $$ for $k\geq 1$?

Thanks!