Find eigenvalues and eigenvectors of the matrix G= A superscript T A and orthonomalise them. Call them v subscript 1, v subscript 2 and v subscript 3 according to the order of h subscript 1> or=, h subscript 2> or = h subscript 3 of eigenvalues of G 

To solve the problem, we will proceed step by step:

### 1. **Constructing Matrix G:**
Given that \( G = A^T A \), let's assume \( A \) is a 3x3 matrix. Therefore, \( G \) will also be a 3x3 matrix. If you have the matrix \( A \) available, please provide it. If not, I'll assume a generic form of \( A \) and compute \( G \).

### 2. **Finding Eigenvalues and Eigenvectors of G:**
Once \( G \) is obtained, we need to find the eigenvalues (\( h_1, h_2, h_3 \)) and the corresponding eigenvectors of \( G \).

### 3. **Orthonormalization:**
After finding the eigenvectors, we'll orthonormalize them. This will result in three orthonormal eigenvectors, which we will call \( v_1 \), \( v_2 \), and \( v_3 \).

### 4. **Ordering the Eigenvectors:**
The eigenvectors will be ordered according to the magnitude of the corresponding eigenvalues such that:
\[
h_1 \geq h_2 \geq h_3
\]
and the corresponding eigenvectors will be labeled as \( v_1 \), \( v_2 \), and \( v_3 \).

Let's go through this process. Could you please provide the matrix \( A \), or would you like to proceed with a generic matrix \( A \)?

Learn how to find eigenvalues and eigenvectors of the matrix G = A^T A and orthonormalize them step-by-step. This problem involves linear algebra concepts including matrix transposition, eigenvalues, eigenvectors, and orthonormalization. Suitable for advanced undergraduate or graduate students.

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