A = [1,0,1;0,1,1]. Find eigenvalues and eigenvectors of the matrix G= A superscript T A, and orthonomalise them. Call them v1,v2 and v3 according to the order of h1> or= h2>or=h3 of eigenvalues of G

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:

\[
v_1 = \begin{bmatrix} -0.4082 \\ -0.4082 \\ -0.8165 \end{bmatrix}, \quad
v_2 = \begin{bmatrix} -0.7071 \\ 0.7071 \\ -2.6124 \times 10^{-16} \end{bmatrix}, \quad
v_3 = \begin{bmatrix} -0.5774 \\ -0.5774 \\ 0.5774 \end{bmatrix}
\]

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.

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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.

Learn how to compute eigenvalues and eigenvectors of matrix G = A^T A using matrix algebra. Understand the steps involved in sorting and orthonormalizing eigenvectors based on their corresponding eigenvalues. Suitable for advanced undergraduate students and those interested in linear algebra.

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