$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$
Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant.
How can write it as in roots of unity.
How to find the eigenvalues of $A$. Please explain.
Use Circulant matrix:
Let $\omega_j = \exp(\frac{2\pi ij}n)$ be the $n$-th roots of unity. Then the vectors $$ v_j = \frac1{\sqrt n} (1,\ \omega_j, \dots, \omega_j^{n-1})^T $$ are eigenvectors of $B$ and $C$ to eigenvalues $\lambda_j^B$ and $\lambda_j^C$, respectively. Set $$ Q= (v_1 \dots v_n) \in\mathbb C^{n,n}. $$ Then $Q$ diagonalizes $B$ and $C$ simultaneously, and it holds $$ \begin{pmatrix} Q^H&0\\0&Q^H\end{pmatrix} A \begin{pmatrix} Q&0\\0&Q\end{pmatrix} = \begin{pmatrix} D_B&D_C\\D_C&D_B\end{pmatrix}, $$ where $D_B$ and $D_C$ are diagonal matrices with the eigenvalues of $B$ and $C$ on the diagonal. It follows that the eigenvalues of $A$ are given as the eigenvalues of $2\times 2$ matrices $$ \begin{pmatrix} \lambda_j^B & \lambda_j^C\\ \lambda_j^C & \lambda_j^B \end{pmatrix}, $$ which are $\lambda_j^B \pm \lambda_j^C$ with corresponding eigenvectors $$ \begin{pmatrix} v_j \\ \pm v_j\end{pmatrix}. $$