Matrix Norms - Burden and Faires(Page 439) Burden and Faires define:

If $\|\cdot\|$ is a vector norm in $\mathbb{R}^n$, then
$$\|A\|=\max_{\|x\|=1}\|Ax\|.$$

In other words, the measure given to a matrix under such a norm describes how the matrix stretches unit vectors of length $1$ ($\|x\|=1$), relative to that norm. The norm of the matrix is the maximum stretch.
They illustrate by means of an example, for the matrix
$$A =\begin{bmatrix}
0 & -2 \\
2 & 0
\end{bmatrix}.$$
Graphically, the matrix norm is:

The image of the vector any vector $x=(x_1,x_2)$ is $Ax=(-2x_2,2x_1)$. When I plugged in some vectors $(1,0),(0,1),(-1,0),(0,-1)$ for the $l_\infty$ norm as well as some others like $(1/\sqrt{2},1/\sqrt{2})$ for the $l_2$ norm, I don't get the parallelogram or the ellipse. Could someone help me visualize this better? I guess I'm missing something.
 A: It is the wrong matrix. One can relatively easily reconstruct a matrix from the first image, as the sides of the unit square map to the sides of the parallelogram. Assume that the orientation is preserved, as it is in the given matrix. From the top and bottom sides as images of $x=\pm 1$, $y\in[-1,1]$, the $y$ component of the image is more like $1.3x$ and in the left side as image of $y=1$, $x\in[-1,1]$ the $x$ component varies from $-2$ at $x=1$ to about $-1.2$ at $x=-1$, giving a formula like $-0.4x-1.6y$, (on the other hand, the intercept with the $x$-axis is at about $-1.5$,...) so that 
$$
A=\begin{bmatrix}-0.4&-1.6\\1.3&0\end{bmatrix}
$$

def plot_A(x,y):
    plt.plot(-(0.3*x+1.5*y),1.3*x); plt.grid();
    plt.xlim(-2.1,2.1); plt.ylim(-1.4,1.4)

plt.figure(figsize=(2*3,2));
x=np.array([1,-1,-1, 1,1]);
y=np.array([1, 1,-1,-1,1]);
plt.subplot(1,2,1); plot_A(x,y);
phi=np.linspace(0,2*np.pi,200);
x,y = np.cos(phi), np.sin(phi);
plt.subplot(1,2,2); plot_A(x,y);
plt.tight_layout(); plt.show();

