In the notation and conventions used in the book by M. P. Deisenroth et al., the matrices

U, 2. V appearing in the singular value decomposition of a matrix A are:

$U = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}}  \\ \frac{2}{\sqrt{5}}  & \frac{1}{\sqrt{5}} \end{bmatrix}$  

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

$V = \begin{bmatrix} \frac{\sqrt{5}}{3} & 0 & 2/3  \\ \frac{2}{3\sqrt{5}}  & \frac{-2}{\sqrt{5}} & \frac{-1}{3} \\ \frac{-4}{3\sqrt{5}}  & \frac{-1}{\sqrt{5}} & \frac{2}{3} \end{bmatrix}$   

Calculate the matrix A.

The singular value decomposition (SVD) of a matrix $ A $ is given by $ A = U S V^T $, where $ U $, $ S $, and $ V $ are the matrices provided:

- $ U = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} $
- $ S = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \end{bmatrix} $
- $ V = \begin{bmatrix} \frac{\sqrt{5}}{3} & 0 & \frac{2}{3} \\ \frac{2}{3\sqrt{5}} & \frac{-2}{\sqrt{5}} & \frac{-1}{3} \\ \frac{-4}{3\sqrt{5}} & \frac{-1}{\sqrt{5}} & \frac{2}{3} \end{bmatrix} $

First, compute the transpose of $ V $, denoted as $ V^T $:

\[
V^T = \begin{bmatrix} 
\frac{\sqrt{5}}{3} & \frac{2}{3\sqrt{5}} & \frac{-4}{3\sqrt{5}} \\
0 & \frac{-2}{\sqrt{5}} & \frac{-1}{\sqrt{5}} \\
\frac{2}{3} & \frac{-1}{3} & \frac{2}{3}
\end{bmatrix}
\]

Next, compute the product $ S V^T $:

\[
S V^T = \begin{bmatrix} 
3 & 0 & 0 \\
0 & 2 & 0 
\end{bmatrix}
\begin{bmatrix} 
\frac{\sqrt{5}}{3} & \frac{2}{3\sqrt{5}} & \frac{-4}{3\sqrt{5}} \\
0 & \frac{-2}{\sqrt{5}} & \frac{-1}{\sqrt{5}} \\
\frac{2}{3} & \frac{-1}{3} & \frac{2}{3}
\end{bmatrix}
\]

This yields:

\[
S V^T = \begin{bmatrix} 
3 \cdot \frac{\sqrt{5}}{3} & 3 \cdot \frac{2}{3\sqrt{5}} & 3 \cdot \frac{-4}{3\sqrt{5}} \\
2 \cdot 0 & 2 \cdot \frac{-2}{\sqrt{5}} & 2 \cdot \frac{-1}{\sqrt{5}}
\end{bmatrix} = \begin{bmatrix} 
\sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\
0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}}
\end{bmatrix}
\]

Finally, compute the product $ A = U (S V^T) $:

\[
A = \begin{bmatrix} 
\frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ 
\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} 
\end{bmatrix}
\begin{bmatrix} 
\sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\
0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}}
\end{bmatrix}
\]

Breaking down the multiplication:

\[
\begin{bmatrix} 
\frac{1}{\sqrt{5}} & \frac{-2}{\sqrt{5}} \\ 
\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} 
\end{bmatrix}
\begin{bmatrix} 
\sqrt{5} & \frac{2}{\sqrt{5}} & \frac{-4}{\sqrt{5}} \\
0 & \frac{-4}{\sqrt{5}} & \frac{-2}{\sqrt{5}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{\sqrt{5}} \cdot \sqrt{5} + \frac{-2}{\sqrt{5}} \cdot 0 & \frac{1}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} + \frac{-2}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} & \frac{1}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} + \frac{-2}{\sqrt{5}} \cdot \frac{-2}{\sqrt{5}} \\ 
\frac{2}{\sqrt{5}} \cdot \sqrt{5} + \frac{1}{\sqrt{5}} \cdot 0 & \frac{2}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} + \frac{1}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} & \frac{2}{\sqrt{5}} \cdot \frac{-4}{\sqrt{5}} + \frac{1}{\sqrt{5}} \cdot \frac{-2}{\sqrt{5}}
\end{bmatrix}
\]

\[
= \begin{bmatrix}
1 & \frac{2}{5} + \frac{8}{5} & \frac{-4}{5} + \frac{4}{5} \\ 
2 & \frac{4}{5} + \frac{-4}{5} & \frac{-8}{5} + \frac{-2}{5}
\end{bmatrix}
\]

\[
= \begin{bmatrix}
1 & 2 & 0 \\ 
2 & 0 & -2
\end{bmatrix}
\]

Therefore, the matrix $ A $ is:

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

Would you like further details or have any questions? Here are some additional questions you might consider:

1. What is the significance of singular value decomposition (SVD) in linear algebra?
2. How do the orthogonal properties of $ U $ and $ V $ affect the SVD of a matrix?
3. Can SVD be used to approximate a matrix with lower rank?
4. How do singular values relate to the eigenvalues of a matrix?
5. What are some practical applications of SVD in data science?

**Tip:** When performing matrix multiplications, double-check each element computation to ensure accuracy.

Learn how to calculate matrix A using Singular Value Decomposition (SVD) with detailed step-by-step computations. This problem covers the SVD process involving matrices U, S, and V, and demonstrates their application in linear algebra. Suitable for undergraduate students studying linear algebra.

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