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 $A$?