# coding: utf-8
# # 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)
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:
# In[3]:
xs = np.random.randn(n, 1000)
# First, we need to bring all those vectors to have norm 1. First, compute the norms:
# In[4]:
p = 2
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.
# In[16]:
normalized_xs = xs/norm_xs
la.norm(normalized_xs[:, 316], p)
# Let's take a look:
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pt.plot(normalized_xs[0], normalized_xs[1], "o")
pt.gca().set_aspect("equal")
# Now apply $A$ to these normalized vectors:
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A_nxs = A.dot(normalized_xs)
# --------------
# Let's take a look again:
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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[9]:
norm_Axs = np.sum(np.abs(A_nxs)**p, axis=0)**(1/p)
norm_Axs.shape
# What's the biggest one?
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np.max(norm_Axs)
# Compare that with what `numpy` thinks the matrix norm is:
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la.norm(A, p)