Find the eigenvalues and eigenvectors of the matrix G = A superscript T A, and orthonomalise them 

To find the eigenvalues and eigenvectors of the matrix \( G = A^T A \) and orthonormalize them, let's follow these steps:

### Step 1: Define Matrix \( G \)

We have already established that:

\[
G = A^T A = 
\begin{bmatrix} 
ax + bp & ay + bq & az + br \\ 
cx + dp & cy + dq & cz + dr \\ 
ex + fp & ey + fq & ez + fr 
\end{bmatrix}
\]

### Step 2: Find Eigenvalues of \( G \)

The eigenvalues \( \lambda \) of matrix \( G \) are found by solving the characteristic equation:

\[
\text{det}(G - \lambda I) = 0
\]

where \( I \) is the identity matrix of the same dimension as \( G \). This step involves calculating the determinant of the matrix \( G - \lambda I \), which results in a cubic equation in \( \lambda \).

### Step 3: Find Eigenvectors of \( G \)

For each eigenvalue \( \lambda \), we solve the system of equations:

\[
(G - \lambda I) \mathbf{v} = 0
\]

where \( \mathbf{v} \) is the eigenvector corresponding to the eigenvalue \( \lambda \). This involves solving a system of linear equations to find the vectors \( \mathbf{v} \).

### Step 4: Orthonormalize the Eigenvectors

Once we have the eigenvectors, we can orthonormalize them using the **Gram-Schmidt process**. The process ensures that the resulting vectors are orthogonal (perpendicular) and have unit length.

1. **Normalize the first eigenvector \( \mathbf{v}_1 \):**

   \[
   \mathbf{u}_1 = \frac{\mathbf{v}_1}{\|\mathbf{v}_1\|}
   \]

2. **Orthogonalize the second eigenvector \( \mathbf{v}_2 \) with respect to \( \mathbf{u}_1 \):**

   \[
   \mathbf{u}_2 = \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2)
   \]

   where 

   \[
   \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) = \frac{\mathbf{v}_2 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1
   \]

   Then normalize \( \mathbf{u}_2 \):

   \[
   \mathbf{u}_2 = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|}
   \]

3. **Repeat the orthogonalization for the third eigenvector \( \mathbf{v}_3 \) with respect to \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):**

   \[
   \mathbf{u}_3 = \mathbf{v}_3 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3)
   \]

   Normalize \( \mathbf{u}_3 \):

   \[
   \mathbf{u}_3 = \frac{\mathbf{u}_3}{\|\mathbf{u}_3\|}
   \]

### Summary

After following these steps, you'll have the orthonormalized eigenvectors of \( G \).

Would you like a detailed walkthrough with specific numerical examples?

### Related Questions:
1. How do you compute the determinant of a \( 3 \times 3 \) matrix?
2. What is the Gram-Schmidt process, and why is it important?
3. How can eigenvalues be used to determine the stability of a system?
4. How do you check if a set of vectors is orthonormal?
5. What is the significance of orthonormal eigenvectors in applications like PCA (Principal Component Analysis)?

**Tip:** Always ensure to verify your eigenvectors' orthogonality after normalization to avoid computational errors.

Learn how to find the eigenvalues and eigenvectors of matrix G = A^T A and orthonormalize them using the Gram-Schmidt process. This problem involves advanced concepts in linear algebra and matrix operations, suitable for students and professionals studying advanced mathematics.

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