**A=A^('). Let \lambda _(1),\lambda _(2) be eigenvalues of A which are unequal and let v_(1),v_(2) be corresponding eigenvectors. Show that v_(1) and v_(2) are orthogonal.**
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Given that \( A = A^\top \), we know that matrix \( A \) is symmetric. A crucial property of symmetric matrices is that their eigenvectors corresponding to distinct eigenvalues are orthogonal.

### Steps to Show Orthogonality

Let \( \lambda_1 \) and \( \lambda_2 \) be eigenvalues of the matrix \( A \), with corresponding eigenvectors \( v_1 \) and \( v_2 \), such that:
\[
A v_1 = \lambda_1 v_1
\]
\[
A v_2 = \lambda_2 v_2
\]
We aim to show that \( v_1 \) and \( v_2 \) are orthogonal, i.e., \( v_1^\top v_2 = 0 \).

### Proof:

1. **Start with the eigenvalue equations:**
   \[
   A v_1 = \lambda_1 v_1 \quad \text{and} \quad A v_2 = \lambda_2 v_2
   \]
   Multiply the first equation on the left by \( v_2^\top \) and the second equation on the left by \( v_1^\top \):
   \[
   v_2^\top A v_1 = \lambda_1 v_2^\top v_1 \quad (1)
   \]
   \[
   v_1^\top A v_2 = \lambda_2 v_1^\top v_2 \quad (2)
   \]

2. **Using symmetry of \( A \):**
   Since \( A = A^\top \), we know that:
   \[
   v_2^\top A v_1 = v_1^\top A v_2
   \]
   Therefore, from equations \( (1) \) and \( (2) \), we have:
   \[
   \lambda_1 v_2^\top v_1 = \lambda_2 v_1^\top v_2
   \]
   This simplifies to:
   \[
   \lambda_1 v_2^\top v_1 = \lambda_2 v_2^\top v_1
   \]

3. **Factor out \( v_2^\top v_1 \):**
   \[
   (\lambda_1 - \lambda_2) v_2^\top v_1 = 0
   \]

4. **Conclusion:**
   Since \( \lambda_1 \neq \lambda_2 \) (by assumption), it must be that:
   \[
   v_2^\top v_1 = 0
   \]
   Hence, \( v_1 \) and \( v_2 \) are orthogonal.

### Final Conclusion:
The eigenvectors \( v_1 \) and \( v_2 \), corresponding to the distinct eigenvalues \( \lambda_1 \) and \( \lambda_2 \) of the symmetric matrix \( A \), are orthogonal.

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Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you compute eigenvalues and eigenvectors for a given matrix?
2. What are some real-world applications of symmetric matrices?
3. Why do symmetric matrices have real eigenvalues?
4. What happens if \( A \) has repeated eigenvalues in terms of eigenvector orthogonality?
5. How does this property extend to complex-valued matrices?

**Tip:** In linear algebra, symmetric matrices are especially important because their eigenvectors form an orthonormal basis, simplifying many computations.

A=A^('). Let λ₁, λ₂ be eigenvalues of A which are unequal and let v₁, v₂ be corresponding eigenvectors. Show that v₁ and v₂ are orthogonal.

Explore how to prove the orthogonality of eigenvectors corresponding to distinct eigenvalues of a symmetric matrix. This solution covers key concepts in linear algebra, including eigenvalues, eigenvectors, and the symmetric matrix theorem. Suitable for undergraduate students studying linear algebra.

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