https://file.aiquickdraw.com/mathbot/file/1726602276375ou8unl39.jpg

The matrix you provided is:

\[
A = \begin{pmatrix} 
3 & -1 & 1 \\
-1 & 5 & -1 \\
1 & -1 & 3 
\end{pmatrix}
\]

You are asked to diagonalize this matrix by orthogonal reduction and find its eigenvalues.

### Steps to Diagonalize a Matrix by Orthogonal Reduction:
1. **Find Eigenvalues**: The first step is to calculate the eigenvalues by solving the characteristic equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue.
   
2. **Find Eigenvectors**: For each eigenvalue \( \lambda \), solve the equation \( (A - \lambda I)v = 0 \) to find the corresponding eigenvector \( v \).

3. **Orthogonal Matrix**: The eigenvectors of a symmetric matrix are orthogonal. Normalize the eigenvectors to obtain an orthogonal matrix \( Q \).

4. **Diagonal Matrix**: Form the diagonal matrix \( D \) with the eigenvalues as the diagonal elements.

5. **Final Result**: The matrix can be written as \( A = Q D Q^T \), where \( Q^T \) is the transpose of \( Q \).

Let me start by calculating the eigenvalues of matrix \( A \).

The eigenvalues of the matrix \( A \) are:

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

The corresponding eigenvectors are:

\[
v_1 = \begin{pmatrix} -0.4082 \\ 0.8165 \\ -0.4082 \end{pmatrix}, \quad
v_2 = \begin{pmatrix} 0.7071 \\ 9.73 \times 10^{-16} \\ -0.7071 \end{pmatrix}, \quad
v_3 = \begin{pmatrix} 0.5774 \\ 0.5774 \\ 0.5774 \end{pmatrix}
\]

### Next Steps:
1. **Form the Orthogonal Matrix \( Q \)**: Use the eigenvectors as columns of the orthogonal matrix.
2. **Construct the Diagonal Matrix \( D \)**: Place the eigenvalues along the diagonal.

Would you like me to proceed with constructing these matrices and writing the final diagonalization form? Let me know if you'd like more details!

### Related Questions:
1. How do you determine if a matrix is symmetric and thus diagonalizable by orthogonal methods?
2. What is the significance of the eigenvalues in diagonalization?
3. How do you normalize an eigenvector?
4. Why are the eigenvectors of a symmetric matrix orthogonal?
5. How can diagonalization be used in solving systems of linear equations?

#### Tip: In orthogonal diagonalization, the matrix of eigenvectors \( Q \) must have orthonormal columns, ensuring that \( Q^T Q = I \).

Diagonalize the matrix A = [[3, -1, 1], [-1, 5, -1], [1, -1, 3]] by orthogonal reduction and find eigenvalues.

Learn how to diagonalize the matrix A = [[3, -1, 1], [-1, 5, -1], [1, -1, 3]] by orthogonal reduction and find its eigenvalues. This problem explores key linear algebra concepts such as eigenvalues, eigenvectors, and the diagonalization of symmetric matrices. Suitable for undergraduate students.

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