diagonalize the matrix orthogonally a= [ 3 -1 1, -1 5 -1, 1 -1 3 ]

To orthogonally diagonalize a matrix \( A \), we need to follow these steps:

1. **Verify if \( A \) is symmetric**: A symmetric matrix can be orthogonally diagonalized, meaning that there exists an orthogonal matrix \( P \) such that \( P^T A P = D \), where \( D \) is a diagonal matrix.
   
   For \( A \):
   \[
   A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix}
   \]
   The matrix is symmetric because \( A^T = A \).

2. **Find the eigenvalues of \( A \)**: To find the eigenvalues, solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \), where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.

3. **Find the eigenvectors of \( A \)**: For each eigenvalue \( \lambda_i \), solve \( (A - \lambda_i I) \mathbf{v} = 0 \) to find the corresponding eigenvectors \( \mathbf{v}_i \).

4. **Construct the orthogonal matrix \( P \)**: Normalize the eigenvectors and arrange them as columns of \( P \). The matrix \( P \) will be orthogonal, meaning \( P^T P = I \).

5. **Diagonal matrix \( D \)**: The diagonal matrix \( D \) will contain the eigenvalues of \( A \) along its diagonal.

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

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

The corresponding eigenvectors (normalized) are:

\[
\mathbf{v}_1 = \begin{bmatrix} -0.4082 \\ 0.8165 \\ -0.4082 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0.7071 \\ 9.7 \times 10^{-16} \\ -0.7071 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 0.5774 \\ 0.5774 \\ 0.5774 \end{bmatrix}
\]

### Orthogonal Matrix \( P \):
The matrix \( P \), whose columns are the normalized eigenvectors, is:

\[
P = \begin{bmatrix}
-0.4082 & 0.7071 & 0.5774 \\
0.8165 & 9.7 \times 10^{-16} & 0.5774 \\
-0.4082 & -0.7071 & 0.5774
\end{bmatrix}
\]

### Diagonal Matrix \( D \):
The diagonal matrix \( D \), with the eigenvalues along the diagonal, is:

\[
D = \begin{bmatrix}
6 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{bmatrix}
\]

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

Would you like more details on any part of this process? Here are a few related questions for deeper understanding:

1. How is the characteristic equation used to find eigenvalues?
2. What does it mean for a matrix to be orthogonally diagonalizable?
3. Why is orthogonality important in diagonalization?
4. How can you verify the orthogonality of the matrix \( P \)?
5. What are some applications of diagonalization in mathematics and physics?

**Tip:** When working with eigenvectors, always check if they are normalized. If not, normalize them to ensure the orthogonal matrix has unit-length columns.

Learn how to orthogonally diagonalize the matrix A = [3 -1 1; -1 5 -1; 1 -1 3] step by step, covering concepts like eigenvalues, eigenvectors, and orthogonality. This detailed guide provides the characteristic equation, eigenvalues, normalized eigenvectors, and final diagonal matrix. Suitable for undergraduate students studying linear algebra.

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