# Vector Norms

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

In [1]:
import numpy as np
import numpy.linalg as la

In [2]:
x = np.array([1.,2,3])


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

In [20]:
np.sum(x**2)**(1/2)

Out[20]:
3.7416573867739413


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

In [21]:
la.norm(x, 2)

Out[21]:
3.7416573867739413


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

Different values of $p$ work similarly:

In [3]:
np.sum(np.abs(x)**5)**(1/5)

Out[3]:
3.0773848853940629

In [23]:
la.norm(x, 5)

Out[23]:
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 [24]:
x**100

Out[24]:
array([  1.00000000e+00,   1.26765060e+30,   5.15377521e+47])

In [25]:
np.sum(x**100)

Out[25]:
5.1537752073201132e+47


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

In [26]:
np.sum(x**100)**(1/100)

Out[26]:
3.0


Numpy can compute that, too:

In [27]:
la.norm(x, np.inf)

Out[27]:
3.0