To find the Singular Value Decomposition (SVD) of matrix \( C = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \), we follow these steps:

The SVD of a matrix \( C \) is given by:

\[
C = U \Sigma V^T
\]

Where:
- \( U \) is an \( m \times m \) orthogonal matrix.
- \( \Sigma \) is an \( m \times n \) diagonal matrix with the singular values.
- \( V \) is an \( n \times n \) orthogonal matrix.

### Step 1: Find \( C^T C \)

\[
C^T = \begin{bmatrix} 0 & 0 & 0 \\ 2 & 0 & 0 \end{bmatrix}
\]

\[
C^T C = \begin{bmatrix} 0 & 0 & 0 \\ 2 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 0 \end{bmatrix}
\]

### Step 2: Find the eigenvalues of \( C^T C \)

The matrix \( C^T C = \begin{bmatrix} 4 & 0 \\ 0 & 0 \end{bmatrix} \) has the eigenvalues 4 and 0. The singular values \( \sigma_1, \sigma_2 \) are the square roots of these eigenvalues:

\[
\sigma_1 = \sqrt{4} = 2, \quad \sigma_2 = \sqrt{0} = 0
\]

So, the diagonal matrix \( \Sigma \) is:

\[
\Sigma = \begin{bmatrix} 2 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}
\]

### Step 3: Find \( V \)

The eigenvectors of \( C^T C \) correspond to the columns of \( V \). The eigenvector for the eigenvalue 4 is \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \), and for the eigenvalue 0 is \( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). Therefore:

\[
V = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

### Step 4: Find \( U \)

We can now compute \( U \) using the formula \( U = C V \Sigma^{-1} \). Applying this:

\[
U = C \cdot V = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}
\]

We normalize the columns to get an orthonormal matrix:

\[
U = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}
\]

Thus, the SVD of \( C \) is:

\[
C = U \Sigma V^T
\]
Where:
\[
U = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}, \quad \Sigma = \begin{bmatrix} 2 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}, \quad V = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

Would you like further details or clarifications?

Here are some related questions to explore:

1. How are singular values related to the eigenvalues of \( C^T C \)?
2. What properties do the matrices \( U \) and \( V \) possess in the SVD?
3. How does SVD differ from Eigenvalue Decomposition?
4. Can you find the SVD of a general 3x3 matrix?
5. How can the SVD be used in image compression?

**Tip:** SVD is a powerful tool used for dimensionality reduction and solving linear systems, especially in machine learning and data science applications.

Explore the Singular Value Decomposition (SVD) of the matrix C = [[0, 2], [0, 0], [0, 0]]. This problem covers key linear algebra concepts, including SVD, matrix decomposition, and eigenvalues. Suitable for undergraduate students studying linear algebra.

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