In [1, p. 34 exercise 2.11] we are asked to find the eigenvalues of the circulant matrix given in [1, (2.27),p.22].
Solution: The matrix given in [1, (2.27),p.22] is
\mathbf{R}=\sigma^2 \mathbf{I}+ P_1 \mathbf{e_1}\mathbf{e}_1^H+ P_2 \mathbf{e}_2\mathbf{e}^H_2 . (1)

According to [1, (2.42),p.27] we can write the matrix as
\mathbf{R}=\sum\limits_{k=0}^{n-1}\rho_k\mathbf{P}^k , (2)

with
\mathbf{P} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & 0 & 0 & \cdots & 0 \\ \end{array}} \right], (3)

\mathbf{P}^0=\mathbf{I} and where \rho_k ,\; k=0,1,2..,n-1 are the elements of the first row of the matrix \mathbf{R}. From the eigenvectors \mathbf{v}_i and eigenvalues \lambda_i of \mathbf{P} we can derive the eigenvalues of \mathbf{R} by:
\mathbf{R}\mathbf{v}_i=\sum\limits_{k=0}^{n-1}\rho_k\mathbf{P}^k \mathbf{v}_i
=\rho_0 \mathbf{v}_i+ \sum\limits_{k=1}^{n-1}\rho_k\mathbf{P}^{k-1}\left(\mathbf{P} \mathbf{v}_i \right)
=\rho_0 \mathbf{v}_i+ \lambda_i\sum\limits_{k=1}^{n-1}\rho_k\mathbf{P}^{k-1}\mathbf{v}_i
=\rho_0 \mathbf{v}_i+ \lambda_i \rho_1 \mathbf{v}_i+\lambda_i\sum\limits_{k=2}^{n-1}\rho_k\mathbf{P}^{k-1}\mathbf{v}_i
=...
=\left(\rho_0+\lambda_i\rho_1+\lambda_i^2 \rho_2+...+\rho_{n-1}\lambda_i^{n-1}\right)\mathbf{v}_i (4)

Thus the eigenvalues of the matrix \mathbf{R} are given by r_i=\rho_0+\lambda_i\rho_1+\lambda_i^2 \rho_2+...+\rho_{n-1}\lambda_i^{n-1} and are related to the eigenvectors \mathbf{v_i}=\frac{1}{\sqrt{n}} \left[ {\begin{array}{*{20}c} 1 & {e^{ - j\frac{{2\pi }}{n}(i - 1)} } & {e^{ - j\frac{{2\pi }}{n}2(i - 1)} } & \cdots & {e^{ - j\frac{{2\pi }}{n}(n - 1)(i - 1)} } \\ \end{array}} \right]^T where \lambda_i=e^{-j\frac{2\pi}{n}(i-1)} with i=1,2,...,n. The elements of the first row of the matrix \mathbf{R} are \rho_0=\sigma^2+P_1+P_2, \rho_1=P_1e^{-j2\pi f_1}+P_2 e^{-j2\pi f_2}, ..., \rho_{n-1}=P_1e^{-j(n-1)2\pi f_1}+P_2 e^{-j(n-1)2\pi f_2} , as can be seen in the solution of [1, p. 34 exercise 2.5] given in [2, solution of exercise 2.5].

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.
[2] Panagiotis Chatzichrisafis: “Solution of exercise 2.5 from Kay’s Modern Spectral Estimation -Theory and Applications”, lysario.de.