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The image appears to show a handwritten note on diagonalizing a matrix using orthogonal transformations. Here's a breakdown of what's shown:

1. **Introduction to Orthogonal Transformations:**
   - An orthogonal transformation transforms a matrix into a diagonal matrix using the relation \( N^T A N \), where \( N \) is an orthogonal matrix. 
   - A matrix \( A \) is orthogonal if \( A^{-1} = A^T \).
   - Properties: \( AA^T = A^T A = I \), where \( I \) is the identity matrix.

2. **Matrix to Diagonalize:**
   \[
   A = \begin{pmatrix}
   3 & 1 & 1 \\
   1 & 3 & -1 \\
   1 & -1 & 3
   \end{pmatrix}
   \]
   This matrix needs to be diagonalized.

3. **Eigenvalues \( \lambda \):**
   - \( \lambda_1 = 1 \), \( \lambda_2 = 4 \), \( \lambda_3 = 4 \).

4. **Eigenvectors:**
   - For \( \lambda_1 = 1 \): \( x_1 = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix} \)
   - For \( \lambda_2 = 4 \): \( x_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \)
   - For \( \lambda_3 = 4 \): \( x_3 = \begin{pmatrix} -1 \\ 1 \\ -2 \end{pmatrix} \)

5. **Diagonalized Matrix:**
   After applying the orthogonal transformation, the resulting diagonal matrix will have the eigenvalues as its diagonal entries.

6. **Additional Matrices:**
   There are additional matrices provided with different eigenvalues (like \( \lambda = 0,1,3 \) and \( \lambda = -1,2,2 \)) that seem to be exercises or homework tasks for further diagonalization by orthogonal transformations.

Would you like a detailed step-by-step explanation of diagonalization, or help with a specific part of this image?

### 5 Related Questions:
1. What is the process to find eigenvalues for a given matrix?
2. How are eigenvectors derived for each eigenvalue?
3. What properties does an orthogonal matrix have?
4. How do you confirm that a matrix is diagonalizable?
5. Can all square matrices be diagonalized using orthogonal transformations?

### Tip:
When diagonalizing matrices, always check if the matrix is symmetric. Symmetric matrices are always diagonalizable using orthogonal matrices, simplifying the process!

How do you diagonalize the given matrix A using orthogonal transformations?

Learn how to diagonalize a matrix using orthogonal transformations with a detailed step-by-step solution. The process includes finding eigenvalues, eigenvectors, and applying the orthogonal diagonalization theorem. Suitable for college-level linear algebra students.

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