Vector Norms

$p$-norms can be computed in two different ways in numpy:

In [1]:
#keep
import numpy as np
import numpy.linalg as la
In [2]:
#keep
x = np.array([1.,2,3])

First, let's compute the 2-norm by hand:

In [3]:
np.sum(x**2)**(1/2)
Out[3]:
1.0

Next, let's use numpy machinery to compute it:

In [4]:
la.norm(x, 2)
Out[4]:
3.7416573867739413

Both of the values above represent the 2-norm: $\|x\|_2$.


Different values of $p$ work similarly:

In [5]:
#keep
np.sum(np.abs(x)**5)**(1/5)
Out[5]:
1.0
In [6]:
la.norm(x, 5)
Out[6]:
3.0773848853940629

The $\infty$ norm represents a special case, because it's actually (in some sense) the limit of $p$-norms as $p\to\infty$.

Recall that: $\|x\|_\infty = \max(|x_1|, |x_2|, |x_3|)$.

Where does that come from? Let's try with $p=100$:

In [7]:
x**100
Out[7]:
array([  1.00000000e+00,   1.26765060e+30,   5.15377521e+47])
In [8]:
np.sum(x**100)
Out[8]:
5.1537752073201132e+47

Compare to last value in vector: the addition has essentially taken the maximum:

In [9]:
np.sum(x**100)**(1/100)
Out[9]:
1.0

Numpy can compute that, too:

In [10]:
la.norm(x, np.inf)
Out[10]:
3.0
In [ ]: