Matrix norms¶

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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:

n = 2

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

A

array([[-0.12603194,  0.74240991],
       [ 0.07209953, -0.91593647]])

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)

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

p = 2

norm_xs = np.sum(np.abs(xs)**p, axis=0)**(1/p)

norm_xs.shape

(1000,)

Then, divide by the norms and assign to normalized_xs:

Then check the norm of a randomly chosen vector.

#clear

normalized_xs = xs/norm_xs

la.norm(normalized_xs[:, 316], p)

1.0

Let's take a look:

pt.plot(normalized_xs[0], normalized_xs[1], "o")

pt.gca().set_aspect("equal")

Now apply $A$ to these normalized vectors:

#clear

A_nxs = A.dot(normalized_xs)

Let's take a look again:

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:

#clear

norm_Axs = np.sum(np.abs(A_nxs)**p, axis=0)**(1/p)

norm_Axs.shape

(1000,)

What's the biggest one?

#clear

np.max(norm_Axs)

1.4175372998707851

Compare that with what numpy thinks the matrix norm is:

#clear

la.norm(A, p)

1.4175373681672399