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:

-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.