understanding a matrix geometrically For fun, I set myself the exercise of understanding the matrix 
 \begin{pmatrix}
  1 & 2\\
  3 & 4 
 \end{pmatrix}
geometrically.
What I am looking for is a decomposition of this matrix into simpler linear transformations like reflections, shears, scalings and rotations.
Things I tried:
Constructing the image of the unit square.
Constructing the image of the axes.
Computing eigenvalues/eigenvectors.
Things I know:
Based on the above, the linear transformation represented by the matrix transforms lines into lines and reverses orientation. Also it is not an isometry.
 A: IMO the singular-value decomposition is a much better way to visualize the geometrical behavior of a matrix, than the eigenvalue decomposition. 
The decomposition for a matrix $M∈ℝ^{m×n}$ is given by:
$$M=UΣV^\top$$
with $U∈ℝ^{m×m}$ orthogonal, $V∈ℝ^{n×n}$ orthogonal and $$Σ=\left(\begin{array}{ccc|ccc}ο_1 & && & \vdots & \\
 &\ddots &&\cdots& 0 & \cdots\\ 
&&σ_r & & \vdots & \\ \hline 
 & \vdots & & & \vdots & \\
 \cdots& 0 & \cdots &  \cdots& 0 & \cdots\\
 & \vdots & & & \vdots & \\
 \end{array}\right)∈ℝ^{m×n}$$
With this decomposition the mapping $$\tilde{M}:ℝ^n→R^m\\x↦Mx$$
can be seen as the composition of three mappings $\tilde{M}=\tilde{U}\tilde{Σ}\tilde{V^\top}$, with 
$$\tilde{V^\top}:ℝ^n→R^n\\x↦V^{\top}x$$
$$\tilde{Σ}:ℝ^n→R^m\\x↦Σx$$
$$\tilde{U}:ℝ^m→R^m\\x↦Ux$$
The mappings $V^\top$ and $U$ define coordinate transformations in $ℝ^n$ / $ℝ^m$, and $Σ$ is a scaling, as it is diagonal. 
You can see that in the following picture, that illustrates the transformation of the unit disc. The arrows represent basis vectors.

Source

For your matrix it is 
$$A=\begin{pmatrix}
  -0.4046  &-0.9145 \\
  -0.9145  & 0.4046
\end{pmatrix}
\begin{pmatrix}   5.4650   \\ &0.3660\end{pmatrix}
\begin{pmatrix}
  -0.5760  & -0.8174 \\
  0.8174  &-0.5760 
\end{pmatrix}$$
A: Based on the comment of  정준환, I found a following decomposition of A into elementary matrices.
$
 \begin{pmatrix}
  1 & 1/2 \\
  0 & 1 \\
  \end{pmatrix}
\begin{pmatrix}
-1/2 & 0 \\
  0 & 1 \\
 \end{pmatrix}
\begin{pmatrix}
1 & 0 \\
3 & 1 \\
 \end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & 4 \\
\end{pmatrix}
= A
$
These correspond to horizontal shear, horizontal scaling, vertical shear and vertical scaling. 
