linalg.norm

Description

Matrix or vector norm.

This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described in Table 1), depending on the value of the ord parameter.

**Table 1** Norms corresponding to values of the **ord** parameter
ord	Norm for Matrices	Norm for Vectors
None	Frobenius norm	2-norm
'fro'	Frobenius norm	-
'nuc'	Nuclear norm	-
inf	max(sum(abs(x), axis=1))	max(abs(x))
-inf	min(sum(abs(x), axis=1))	min(abs(x))
0	-	sum(x != 0)
1	max(sum(abs(x), axis=0))	sum(abs(x)ord)(1./ord)
-1	min(sum(abs(x), axis=0))	sum(abs(x)ord)(1./ord)
2	2-norm (largest singular value)	sum(abs(x)ord)(1./ord)
-2	Smallest singular value	sum(abs(x)ord)(1./ord)
Other	-	sum(abs(x)ord)(1./ord)

Mandatory Input Parameters

Parameter	Type	Description
x	array_like	Input array If axis is None, x must be 1D or 2D, unless ord is None. If both axis and ord are None, the 2-norm of x.ravel will be returned.

Parameter

Type

Description

array_like

Input array

If axis is None, x must be 1D or 2D, unless ord is None.
If both axis and ord are None, the 2-norm of x.ravel will be returned.

Optional Input Parameters

Parameter	Type	Default Value	Description
ord	{non-zero int, inf, -inf, 'fro', 'nuc'}	None	For details, see Table 1.
axis	{None, int, 2-tuple of ints}	None	If axis is an integer, it specifies the axis of x along which to compute the vector norms. If axis is a 2-tuple, it specifies the axes that hold 2D matrices, and the matrix norms of these matrices are computed. If axis is None, then either a vector norm (when x is 1D) or a matrix norm (when x is 2D) is returned.
keepdims	bool	False	If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original x.

Return Value

Type	Description
float or ndarray	Matrix or vector norm.

Examples

>>> import numpy as np
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1,  0,  1,  2,  3,  4])
>>> b = a.reshape((3,3))
>>> b
array([[-4, -3, -2],
       [-1,  0,  1],
       [ 2,  3,  4]])
>>> 
>>> np.linalg.norm(a)
7.745966692414834
>>> np.linalg.norm(b)
7.745966692414834
>>> np.linalg.norm(b, 'fro')
7.745966692414834
>>> np.linalg.norm(a, np.inf)
4.0
>>> np.linalg.norm(b, np.inf)
9.0
>>> np.linalg.norm(a, -np.inf)
0.0
>>> np.linalg.norm(b, -np.inf)
2.0
>>> 
>>> np.linalg.norm(a, 1)
20.0
>>> np.linalg.norm(b, 1)
7.0
>>> np.linalg.norm(b, -1)
6.0
>>> np.linalg.norm(a, 2)
7.745966692414834
>>> np.linalg.norm(b, 2)
7.348469228349533
>>> np.linalg.norm(a, -2)
0.0
>>> np.linalg.norm(b, -2)
3.757577037301703e-16
>>> 
>>> c = np.array([[1,2,3], [-1,1,4]])
>>> np.linalg.norm(c, axis=0)
array([1.41421356, 2.23606798, 5.        ])
>>> np.linalg.norm(c, axis=1)
array([3.74165739, 4.24264069])
>>> np.linalg.norm(c, ord=1, axis=1)
array([6., 6.])
>>> 
>>> 
>>> m = np.arange(8).reshape(2,2,2)
>>> m
array([[[0, 1],
        [2, 3]],

       [[4, 5],
        [6, 7]]])
>>> np.linalg.norm(m, axis=(1,2))
array([ 3.74165739, 11.22497216])
>>> np.linalg.norm(m[0, :, :])
3.7416573867739413
>>> np.linalg.norm(m[1, :, :])
11.224972160321824
>>>

Parent topic: Linear Algebra Functions