The given matrix is:

$A = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}$

We need to find the Singular Value Decomposition (SVD) of this matrix $A$. The SVD of a matrix $A$ is a factorization of the form:

$A = U \Sigma V^T$

Where:

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

I will now compute the SVD of this matrix.The Singular Value Decomposition (SVD) of the matrix $A$ is as follows:

$A = U \Sigma V^T$

Where:

• The left singular vectors $U$ are:

$U = \begin{bmatrix} -0.7071 & 0 & 0.7071 \\ 0 & 1 & 0 \\ 0.7071 & 0 & 0.7071 \end{bmatrix}$

• The singular values $\Sigma$ are:

$\Sigma = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

• The right singular vectors $V^T$ are:

$V^T = \begin{bmatrix} -0.7071 & 0 & 0.7071 \\ 0 & 1 & 0 \\ 0.7071 & 0 & 0.7071 \end{bmatrix}$

This decomposition shows that the matrix can be expressed as the product of these three matrices.

Would you like further details on how SVD is applied or any other part of this calculation?

Here are some related questions to explore further:

1. How are singular values used to measure matrix properties?
2. What do the left and right singular vectors represent geometrically?
3. How does the SVD relate to the eigenvalue decomposition for symmetric matrices?
4. Can the SVD be used for matrix approximation and why?
5. How does the SVD help with solving systems of linear equations?

Tip: SVD is particularly useful for applications like dimensionality reduction (PCA) and solving ill-conditioned systems in numerical analysis.