The matrix $A = \begin{pmatrix} 3 & 0 \\ 0 & -2 \end{pmatrix}$ is a simple 2x2 diagonal matrix, and finding its Singular Value Decomposition (SVD) is straightforward.

The SVD of a matrix $A$ is given by: $A = U \Sigma V^T$ where:

• $U$ is an orthogonal matrix (left singular vectors),
• $\Sigma$ is a diagonal matrix with non-negative singular values,
• $V^T$ is the transpose of an orthogonal matrix (right singular vectors).

Steps to find the SVD:

1. Find the singular values: The singular values of $A$ are the square roots of the eigenvalues of $A^T A$. Since $A$ is diagonal, the eigenvalues are the squares of the diagonal elements of $A$: $A^T A = \begin{pmatrix} 9 & 0 \\ 0 & 4 \end{pmatrix}$ So, the singular values are: $\sigma_1 = 3, \quad \sigma_2 = 2$ Hence, $\Sigma = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$.

2. Find the right singular vectors (matrix $V$): The columns of $V$ are the eigenvectors of $A^T A$. Since $A^T A$ is diagonal, the eigenvectors are just the standard basis vectors: $V = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ Thus, $V^T = V = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

3. Find the left singular vectors (matrix $U$): The columns of $U$ are the eigenvectors of $A A^T$. Since $A A^T = A^T A$, the eigenvectors of both matrices are the same, and $U$ is also the identity matrix: $U = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ (Note that the sign of $U$ can vary depending on the convention used, but this is one possible set of eigenvectors.)

Final SVD:

$A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ or simply: $A = U \Sigma V^T$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you compute the SVD for non-diagonal matrices?
2. What is the geometric interpretation of singular values and vectors?
3. How can SVD be applied to image compression?
4. How does the SVD relate to the eigenvalue decomposition?
5. What is the rank of a matrix and how is it related to SVD?

Tip: The singular values give important insights into the matrix, such as its condition number, which is essential in numerical stability analysis.