Note that A(Cn) = Qn + Q
n
= Qn + Qn
n
−1 is a polynomial in Qn. Thus, the
eigenvalues of A(Cn) are obtained by evaluating the same polynomial at each of the
eigenvalues of Qn. Thus, by Lemma 3.5, the eigenvalues of A(Cn) are ωk + ωn−k ,
k = 1,..., n. Note that
ωk + ωn−k = ω
k
+ ω−k
= e
2
πik
n +
e
−
2πik
n
= 2 cos
2π
k
n ,
k = 1,..., n, and the proof is complete.

The passage you provided appears to be part of a proof regarding the eigenvalues of a specific matrix \( A(C_n) \), which is a function of a circulant matrix \( C_n \). Let’s break it down step by step:

### Understanding the Setup

1. **Matrix \( C_n \):** This is a circulant matrix, which means every row is a cyclic permutation of the first row. The eigenvalues of \( C_n \) have already been established as \( \lambda_k = 2 \cos \left(\frac{2\pi k}{n}\right) \) for \( k = 1, 2, \dots, n \).

2. **Matrix \( A(C_n) \):** This matrix is expressed as \( A(C_n) = Q_n + Q_n^{-1} \), where \( Q_n \) is a matrix associated with \( C_n \) (likely a shift or rotation matrix). The eigenvalues of \( A(C_n) \) are determined by evaluating a polynomial in \( Q_n \) at each of the eigenvalues of \( Q_n \).

### Eigenvalues of \( A(C_n) \)

3. **Eigenvalue Calculation:** By Lemma 3.5 (presumably from the same source), the eigenvalues of \( A(C_n) \) are given by \( \omega_k + \omega_{n-k} \), where \( \omega_k \) are the eigenvalues of \( Q_n \).

4. **Expression for Eigenvalues:** The eigenvalues \( \omega_k + \omega_{n-k} \) are further simplified:
   \[
   \omega_k + \omega_{n-k} = \omega_k + \omega_{-k} = e^{\frac{2\pi i k}{n}} + e^{-\frac{2\pi i k}{n}} = 2 \cos \left(\frac{2\pi k}{n}\right)
   \]
   This matches the previously derived eigenvalues of \( C_n \).

### Conclusion

The proof shows that the eigenvalues of \( A(C_n) \) are indeed \( 2 \cos \left(\frac{2\pi k}{n}\right) \), for \( k = 1, 2, \dots, n \), completing the argument. The key insight is that the eigenvalues of a polynomial in a matrix \( C_n \) can be obtained by applying the same polynomial to the eigenvalues of \( C_n \), which in this case leads directly to the cosine terms.

### Follow-Up Questions

1. How does Lemma 3.5 specifically relate to the eigenvalues of circulant matrices?
2. What is the significance of the matrix \( Q_n \) in the context of circulant matrices?
3. Can this method of deriving eigenvalues be applied to other forms of polynomials in matrices?
4. How does the relationship \( \omega_k + \omega_{n-k} = 2 \cos \left(\frac{2\pi k}{n}\right) \) arise naturally in the context of Fourier analysis?
5. What are the practical applications of these eigenvalue calculations in areas like signal processing or physics?

**Tip:** When dealing with circulant matrices, always consider how their inherent symmetry can simplify the computation of eigenvalues, often involving trigonometric identities or Fourier transforms.

Explore the derivation and significance of eigenvalues of matrix A(C_n), a function of circulant matrix C_n. Learn about circulant matrices, polynomial evaluations, and the application of Lemma 3.5 in mathematical proofs.

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