Function/Matrix norms

In [1]: 
import numpy as np
import numpy.linalg as la

Here's a matrix of which we're trying to compute the norm:

In [2]: 
n = 3

A = np.random.randn(n, n)

Recall:

$$||A||=\max_{x\ne 0} \frac{\|Ax\|}{\|x\|},$$

where the vector norm must be specified, and the value of the matrix norm $\|A\|$ depends on the choice of vector norm.

For instance, for the $p$-norms, we often write:

$$||A||_2=\max_{x\ne 0} \frac{\|Ax\|_2}{\|x\|_2},$$

and similarly for different values of $p$.

We can approximate this by just producing very many random vectors and evaluating the formula:

In [4]: 
xs = np.random.randn(1000, 3)
In [8]: 
Axs = np.einsum("ij,kj->ki", A, xs)
In [9]: 
p = 2
norm_xs = np.sum(np.abs(xs)**p, axis=1)**(1/p)
norm_xs.shape
Out[9]:
(1000,)
In [10]: 
norm_Axs = np.sum(np.abs(Axs)**p, axis=1)**(1/p)
norm_Axs.shape
Out[10]:
(1000,)
In [11]: 
np.max(norm_Axs/norm_xs)
Out[11]:
3.1170636628682233
In [12]: 
la.norm(A, p)
Out[12]:
3.117921601396644