For n ≥ 2, the eigenvalues of Cn are 2 cos 2π
k
n
, k = 1,..., n

For the cyclic matrix \( C_n \), the eigenvalues are given by the expression:

\[
\lambda_k = 2 \cos \left(\frac{2\pi k}{n}\right), \quad k = 1, 2, \dots, n
\]

This result is derived from the structure of the cyclic matrix \( C_n \), which is an \( n \times n \) circulant matrix. The general form of a circulant matrix \( C_n \) is:

\[
C_n = \begin{pmatrix}
c_0 & c_{n-1} & c_{n-2} & \cdots & c_1 \\
c_1 & c_0 & c_{n-1} & \cdots & c_2 \\
c_2 & c_1 & c_0 & \cdots & c_3 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
c_{n-1} & c_{n-2} & c_{n-3} & \cdots & c_0
\end{pmatrix}
\]

For the specific case where \( C_n \) corresponds to the cyclic permutation matrix, each eigenvalue \( \lambda_k \) is determined by the formula given above. Here’s the breakdown:

- **Cyclic matrix definition**: \( C_n \) shifts each row vector one position to the right, with elements wrapping around.
- **Eigenvalues**: The eigenvalues of any circulant matrix are obtained using the Discrete Fourier Transform (DFT) of the first row of the matrix.
- **Formula**: For the permutation matrix, the first row has \( 1 \) in the first position and \( 0 \) elsewhere. The DFT of this row results in the eigenvalues given by the formula \( \lambda_k = 2 \cos \left(\frac{2\pi k}{n}\right) \).

This formula applies for \( k = 1, 2, \dots, n \) and is valid for \( n \geq 2 \).

Would you like more details or have any questions?

Here are some follow-up questions:

1. What is the significance of the eigenvalues in the context of the cyclic matrix \( C_n \)?
2. How are circulant matrices generally useful in linear algebra?
3. Can you derive the eigenvalues of a circulant matrix using the Discrete Fourier Transform?
4. How does the structure of \( C_n \) influence the simplicity of calculating its eigenvalues?
5. What is the relationship between the cyclic matrix and Fourier series?

**Tip:** Understanding the Discrete Fourier Transform (DFT) is crucial for analyzing circulant matrices and their eigenvalues.

Understand the eigenvalues of the cyclic matrix C_n using linear algebra concepts. Learn about circulant matrices and how the Discrete Fourier Transform (DFT) helps derive eigenvalues. Explore the formula \( \lambda_k = 2 \cos \left(\frac{2\pi k}{n}\right) \) for n ≥ 2. Suitable for advanced undergraduate studies.

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