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To diagonalize the given matrix using an orthogonal transformation, we will follow these steps:

Given matrix \( A = \begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix} \), we need to find an orthogonal matrix \( P \) and a diagonal matrix \( D \) such that:

\[
P^T A P = D
\]

Where \( D \) is a diagonal matrix containing the eigenvalues of \( A \), and \( P \) contains the orthonormal eigenvectors of \( A \).

### Steps:

1. **Find the eigenvalues of the matrix \( A \):**
   Solve the characteristic equation:

   \[
   \det(A - \lambda I) = 0
   \]

   Where \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix.

2. **Find the eigenvectors corresponding to each eigenvalue:**
   For each eigenvalue \( \lambda_i \), solve the equation:

   \[
   (A - \lambda_i I) \mathbf{v} = 0
   \]

   to find the corresponding eigenvector \( \mathbf{v}_i \).

3. **Normalize the eigenvectors:**
   Normalize the eigenvectors to form an orthonormal set. These eigenvectors form the columns of the orthogonal matrix \( P \).

4. **Construct the diagonal matrix \( D \):**
   The diagonal matrix \( D \) will have the eigenvalues on its diagonal.

Let me calculate the eigenvalues and eigenvectors for this matrix.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 6, \quad \lambda_2 = 2, \quad \lambda_3 = 3
\]

The corresponding eigenvectors (normalized) are:

\[
v_1 = \begin{pmatrix} -0.4082 \\ 0.8165 \\ -0.4082 \end{pmatrix}, \quad
v_2 = \begin{pmatrix} 0.7071 \\ 0 \\ -0.7071 \end{pmatrix}, \quad
v_3 = \begin{pmatrix} 0.5774 \\ 0.5774 \\ 0.5774 \end{pmatrix}
\]

These eigenvectors form the columns of the orthogonal matrix \( P \), and the eigenvalues form the diagonal elements of the diagonal matrix \( D \).

### Orthogonal matrix \( P \):
\[
P = \begin{pmatrix} 
-0.4082 & 0.7071 & 0.5774 \\
0.8165 & 0 & 0.5774 \\
-0.4082 & -0.7071 & 0.5774 
\end{pmatrix}
\]

### Diagonal matrix \( D \):
\[
D = \begin{pmatrix} 
6 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 
\end{pmatrix}
\]

Thus, the matrix \( A \) is diagonalized as \( P^T A P = D \).

Do you need further details or clarifications?

Here are 5 related questions for further exploration:

1. How do we verify that \( P^T A P = D \)?
2. What is the significance of the eigenvalues in diagonalization?
3. Can all symmetric matrices be diagonalized by an orthogonal transformation?
4. How does the process differ for non-symmetric matrices?
5. What applications does diagonalization have in physical systems?

**Tip**: Eigenvalues indicate important properties like stability and frequency in various applications, especially in physics and engineering.

Diagonalize the matrix (3 -1 1, -1 5 -1, 1 -1 3) by orthogonal transformation.

Learn how to diagonalize the matrix (3, -1, 1; -1, 5, -1; 1, -1, 3) using an orthogonal transformation. This guide covers key concepts such as eigenvalues, eigenvectors, and the spectral theorem, providing a detailed step-by-step solution. Suitable for university-level students in linear algebra.

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