To find the singular value decomposition (SVD) of the given matrix:

$A = \begin{pmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix}$

Steps for SVD:

1. Form the matrix $A^T A$:
   Calculate $A^T$, the transpose of matrix $A$, and then compute $A^T A$.

2. Compute eigenvalues and eigenvectors of $A^T A$. The singular values are the square roots of the eigenvalues of $A^T A$.

3. Compute $U$:
   The matrix $U$ is obtained from the eigenvectors of $AA^T$ and gives the left singular vectors.

4. Compute $V$:
   The matrix $V$ is obtained from the eigenvectors of $A^T A$ and gives the right singular vectors.

Let me calculate this step by step.The singular value decomposition (SVD) of matrix $A = \begin{pmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix}$ is given by:

$A = U \Sigma V^T$

Where:

1. Matrix $U$ (left singular vectors): 0.7071 & -0.7071 \\ 0.7071 & 0.7071 \end{pmatrix}$$
2. Matrix $\Sigma$ (singular values along diagonal): 5 & 0 & 0 \\ 0 & 6.01 \times 10^{-8} & 0 \end{pmatrix}$$
3. Matrix $V$ (right singular vectors): -0.7071 & -0.6667 & 0.2357 \\ -0.7071 & 0.6667 & -0.2357 \\ 0 & 0.3333 & 0.9428 \end{pmatrix}$$

This gives the full SVD decomposition.

Would you like more details or have any questions?

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Here are five related questions to explore further:

1. How do the eigenvalues of $A^T A$ relate to the singular values?
2. Why are the singular values in $\Sigma$ non-negative?
3. How is the rank of a matrix determined from its singular values?
4. What is the physical interpretation of SVD in data analysis?
5. How does SVD help in matrix compression and dimensionality reduction?

Tip: The smallest singular values can often be ignored in practice to reduce noise in data applications.