Lemma 3.5 For n ≥ 2, the eigenvalues of Qn are 1,ω,ω2,...,ωn−1, where
ω = e
2π
i
n
, is the primitive nth root of unity.
Proof The characteristic polynomial of Qn is det(Qn − λ
I) = (−1)n(λn − 1
).
Clearly, the roots of the characteristic polynomial are the
n roots of unity.

**Lemma 3.5:**

For \( n \geq 2 \), the eigenvalues of the matrix \( Q_n \) are \( 1, \omega, \omega^2, \dots, \omega^{n-1} \), where \( \omega = e^{\frac{2\pi i}{n}} \) is the primitive \( n \)th root of unity.

### Proof:

1. **Characteristic Polynomial of \( Q_n \):**

   The characteristic polynomial of \( Q_n \) is given by:

   \[
   \text{det}(Q_n - \lambda I) = (-1)^n (\lambda^n - 1)
   \]

   Here:
   - \( \lambda \) represents the eigenvalues of the matrix \( Q_n \).
   - \( I \) is the identity matrix of the same dimension as \( Q_n \).
   - The determinant of the matrix \( Q_n - \lambda I \) gives the characteristic polynomial.

2. **Roots of the Characteristic Polynomial:**

   The characteristic polynomial \( \lambda^n - 1 \) has \( n \) roots. These roots are the \( n \)th roots of unity, which are:

   \[
   1, \omega, \omega^2, \dots, \omega^{n-1}
   \]

   where \( \omega = e^{\frac{2\pi i}{n}} \) is the primitive \( n \)th root of unity.

3. **Conclusion:**

   Since the roots of the characteristic polynomial correspond to the eigenvalues of the matrix \( Q_n \), the eigenvalues of \( Q_n \) are precisely \( 1, \omega, \omega^2, \dots, \omega^{n-1} \).

### Summary:

The lemma and its proof show that the eigenvalues of the matrix \( Q_n \) are the \( n \)th roots of unity, which include 1 and the complex numbers \( \omega, \omega^2, \dots, \omega^{n-1} \), where \( \omega \) is the primitive \( n \)th root of unity.

### Further Questions:

1. What are some specific examples of matrices \( Q_n \) that have these eigenvalues?
2. How does the matrix \( Q_n \) relate to Fourier transforms or circulant matrices?
3. Can the lemma be extended to other types of matrices or higher dimensions?
4. How do the eigenvalues \( 1, \omega, \omega^2, \dots, \omega^{n-1} \) affect the behavior of \( Q_n \) in applications?
5. What is the significance of the determinant being \( (-1)^n \)? How does it relate to the symmetry of the matrix \( Q_n \)?

### Tip:

When dealing with eigenvalues and characteristic polynomials, always check how the roots of the polynomial relate to the matrix's properties. Understanding these connections is crucial in linear algebra.

Explore Lemma 3.5 which states that for n ≥ 2, the eigenvalues of the matrix Q_n are 1, ω, ω^2, ..., ω^(n-1), where ω = e^(2πi/n) is the primitive nth root of unity. Understand the proof involving the characteristic polynomial and its roots. Suitable for advanced undergraduate students and beyond.

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