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.