import numpy as np
import numpy.linalg as la
import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline
# https://matplotlib.org/users/customizing.html
# print(plt.style.available) # uncomment to print all styles
import seaborn as sns
sns.set(font_scale=2)
plt.style.use('seaborn-whitegrid')
mpl.rcParams['figure.figsize'] = (10.0, 8.0)
Here's a matrix of which we're trying to compute the norm:
n = 2
A = np.random.randn(n, n)
A
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:
xs = np.random.randn(n, 1000)
xs.shape
First, we need to bring all those vectors to have norm 1. First, compute the norms:
p = 1
norm_xs = np.sum(np.abs(xs)**p, axis=0)**(1/p)
norm_xs.shape
Then, divide by the norms and assign to normalized_xs:
Then check the norm of a randomly chosen vector.
$${\rm normalized\_xs}= \frac{x}{||x||_p}$$
normalized_xs = xs/norm_xs
la.norm(normalized_xs[:, 99], p)
Let's take a look at all normalized_xs vectors
plt.plot(normalized_xs[0], normalized_xs[1], "b.")
plt.gca().set_aspect("equal")
Now apply $A$ to these normalized vectors:
$${\rm A\_nxs}= A\frac{x}{||x||_p}$$
A_nxs = A.dot(normalized_xs)
Let's take a look again:
plt.plot(normalized_xs[0], normalized_xs[1], "b.", label="x")
plt.plot(A_nxs[0], A_nxs[1], "r.", label="Ax")
plt.legend()
plt.gca().set_aspect("equal")
Next, compute norms of the $Ax$ vectors:
$${\rm norm\_Axs}= ||Ax||_p$$
norm_Axs = np.sum(np.abs(A_nxs)**p, axis=0)**(1/p)
norm_Axs.shape
What's the biggest one?
np.max(norm_Axs)
Compare that with what numpy thinks the matrix norm is:
la.norm(A, p)