Approximate matrix by a rank 2 matrix using singular values I only understand the singular value decomposition process. Do I apply it to this matrix? 
\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}
What is the idea behind matrix approximation using this process?
Applying SVD, we have $\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}^T\begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^2 & 0 \\ 0&0&\pi^2 \end{bmatrix}$
How to proceed?

Decomposition
By agreement, the SVD orders the singular values. The decomposition here is
$$
  \mathbf{A}
=
%
\left(
\begin{array}{ccc}
 0 & 0 & \pi  \\
 0 & e & 0 \\
 1 & 0 & 0 \\
\end{array}
\right)
%
 = \mathbf{U} \, \Sigma \, \mathbf{V}^{*} =
%
\left(
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1 \\
\end{array}
\right)
%
\left(
\begin{array}{ccc}
 \pi  & 0 & 0 \\
 0 & e & 0 \\
 0 & 0 & 1 \\
\end{array}
\right)
%
\left(
\begin{array}{ccc}
 0 & 0 & 1 \\
 0 & 1 & 0 \\
 1 & 0 & 0 \\
\end{array}
\right).
%
$$
The approximation through order $k$ uses the first $k$ singular values. In your case, $k=2$ and the approximation is
$$
  \mathbf{A}_{2}
=
%
\left(
\begin{array}{ccc}
 0 & 0 & \pi  \\
 0 & e & 0 \\
 0 & 0 & 0 \\
\end{array}
\right)
%
 = \mathbf{U} \, \Sigma \, \mathbf{V}^{*} =
%
\left(
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1 \\
\end{array}
\right)
%
\left(
\begin{array}{ccc}
 \pi  & 0 & 0 \\
 0 & e & 0 \\
 0 & 0 & \color{red}{0} \\
\end{array}
\right)
%
\left(
\begin{array}{ccc}
 0 & 0 & 1 \\
 0 & 1 & 0 \\
 1 & 0 & 0 \\
\end{array}
\right).
%
$$
The red $0$ shows that you have discarded $\sigma_{3}$.
