# Matrix norms

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


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

In [14]:
n = 2
A = np.random.randn(n, n)


Recall:

$$||A||=\max_{\|x\|=1} \|Ax\|,$$

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:

$$||A||_2=\max_{\|x\|=1} \|Ax\|_2,$$

and similarly for different values of $p$.

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

In [16]:
xs = np.random.randn(n, 1000)


First, we need to bring all those vectors to have norm 1. First, compute the norms:

In [32]:
p = 2
norm_xs = np.sum(np.abs(xs)**p, axis=0)**(1/p)
norm_xs.shape

Out[32]:
(1000,)


Then, divide by the norms and assign to normalized_xs:

In [33]:
normalized_xs = xs/norm_xs
la.norm(normalized_xs[:, 316], p)

Out[33]:
1.0


Let's take a look:

In [34]:
pt.plot(normalized_xs[0], normalized_xs[1], "o")
pt.gca().set_aspect("equal")


Now apply $A$ to these normalized vectors:

In [35]:
A_nxs = A.dot(normalized_xs)


Let's take a look again:

In [36]:
pt.plot(normalized_xs[0], normalized_xs[1], "o", label="x")
pt.plot(A_nxs[0], A_nxs[1], "o", label="Ax")
pt.legend()
pt.gca().set_aspect("equal")


Next, compute norms of the $Ax$ vectors:

In []:
norm_Axs = np.sum(np.abs(A_nxs)**p, axis=0)**(1/p)
norm_Axs.shape


What's the biggest one?

In [23]:
np.max(norm_Axs)

Out[23]:
2.3888099889735472


Compare that with what numpy thinks the matrix norm is:

In [24]:
la.norm(A, p)

Out[24]:
2.3888099931809323

In [9]:
A = np.arange(9).reshape(3,3)
A

Out[9]:
array([[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])

In [13]:
np.sum(A)

Out[13]:
36

In []: