To solve the problem, we'll go through the following steps:

1. Compute the matrix $G = A^T A$.
2. Find the eigenvalues and eigenvectors of $G$.
3. Sort the eigenvectors by the magnitude of their corresponding eigenvalues.
4. Orthonormalize the eigenvectors.

Let's start with step 1.

Step 1: Compute the matrix $G = A^T A$

Given the matrix $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$,

First, calculate the transpose of $A$, which is $A^T$: $A^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix}$

Now, compute $G = A^T A$: $G = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}$

Step 2: Find the eigenvalues and eigenvectors of $G$

The eigenvalues $\lambda$ are found by solving the characteristic equation: $\det(G - \lambda I) = 0$ Where $I$ is the identity matrix.

Step 3: Sort the eigenvectors by the magnitude of their corresponding eigenvalues

Step 4: Orthonormalize the eigenvectors

Let me compute these steps for you.The matrix $G$ is:

$G = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}$

Next, we'll compute the eigenvalues and eigenvectors of $G$.The eigenvalues of the matrix $G$ are:

$\lambda_1 = 3, \quad \lambda_2 = 1, \quad \lambda_3 = -3.37 \times 10^{-17} \ (\text{approximately 0})$

The corresponding eigenvectors (before orthonormalization) are:

v_2 = \begin{bmatrix} -0.7071 \\ 0.7071 \\ -2.6124 \times 10^{-16} \end{bmatrix}, \quad v_3 = \begin{bmatrix} -0.5774 \\ -0.5774 \\ 0.5774 \end{bmatrix}$$ Finally, let's orthonormalize these eigenvectors to ensure they form an orthonormal set.The orthonormalized eigenvectors corresponding to the eigenvalues $$\lambda_1 = 3$$, $$\lambda_2 = 1$$, and $$\lambda_3 \approx 0$$ are: $$v_1 = \begin{bmatrix} 0.8911 \\ 0.1980 \\ -0.4082 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -0.1715 \\ -0.6860 \\ -0.7071 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -0.4201 \\ 0.7001 \\ -0.5774 \end{bmatrix}$$ These vectors are sorted according to the order of the eigenvalues, and they form an orthonormal set. --- Do you want more details on any of the steps or have any other questions? Here are 5 related questions you might find interesting: 1. How can we interpret the eigenvalues and eigenvectors geometrically? 2. What are the implications if an eigenvalue is zero? 3. How does the Gram-Schmidt process work for orthonormalization? 4. Can the eigenvectors form a basis for the entire vector space? 5. How does the orthonormalization affect the properties of the eigenvectors? **Tip:** The magnitude of the eigenvalue gives an indication of the stretching factor along the direction of its corresponding eigenvector.