# Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$

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.

• Add a dollar sign before "A=" and after "\right)" – tpb261 Jun 24 '14 at 5:31
• Thank you Sir. I have modified. – G Velmurugan Jun 24 '14 at 5:34

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 $$\begin{pmatrix} v_j \\ \pm v_j\end{pmatrix}.$$