ELemma 3.5 n ≥ 2 க்கு, On இன் ஈஜென் மதிப்புகள் 1, w, w²,..., ω"-1 ஆகும், இதில் 2πί நாம், ஒற்றுமையின் பழமையான n வது வேர் ஆகும்.

Given the context, the question seems to be related to a mathematical concept involving the eigenvalues of a matrix \( O_n \) for \( n \geq 2 \). Here's a breakdown of the content:

**Lemma 3.5:**

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

### Explanation:

1. **\( O_n \)**: This is likely a specific matrix related to an \( n \)-dimensional object, possibly the Fourier matrix or another circulant matrix, which is known to have such eigenvalues.
  
2. **Eigenvalues**: The values \( 1, \omega, \omega^2, \dots, \omega^{n-1} \) are the distinct eigenvalues of \( O_n \), which corresponds to the powers of the \( n \)th root of unity \( \omega \).
  
3. **Primitive \( n \)th Root of Unity**: \( \omega = e^{\frac{2\pi i}{n}} \) is a complex number which, when raised to the power \( n \), equals 1 but does not equal 1 for any smaller positive integer power, hence "primitive."

### Questions for Further Exploration:

1. What are the properties of the matrix \( O_n \)?
2. How are the eigenvalues of \( O_n \) derived?
3. What role do the roots of unity play in the structure of \( O_n \)?
4. Can you explain the significance of each eigenvalue \( \omega^k \) in the context of \( O_n \)?
5. How is the lemma applied in a broader mathematical context?

### Tip:

Understanding the concept of roots of unity and their properties can significantly enhance your ability to work with complex eigenvalues in linear algebra.

Explore the mathematical concept of eigenvalues and roots of unity in circulant matrices. Learn about Lemma 3.5, which discusses the eigenvalues of a circulant matrix \( O_n \) for \( n \geq 2 \). Discover the significance of primitive \( n \)th roots of unity in linear algebra.

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