# 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) # 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 # Then, divide by the norms and assign to `normalized_xs`: # In[33]: normalized_xs = xs/norm_xs la.norm(normalized_xs[:, 316], p) # 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) # Compare that with what `numpy` thinks the matrix norm is: # In[24]: la.norm(A, p) # In[9]: A = np.arange(9).reshape(3,3) A # In[13]: np.sum(A) # In[ ]: