The numpy.linalg.norm() method calculates the matrix or vector norm of an input array. Norms quantify the “size” or “magnitude” of vectors and matrices.

A vector norm measures the “length” or “magnitude” of a vector, while a matrix norm measures the size of a matrix in terms of rows, columns, or singular values.

Let’s calculate the vector norm (Euclidean distance L2). It measures straight-line distance.

import numpy as np

vector = np.array([2, 4])

vector_norm = np.linalg.norm(vector)

print(vector_norm)

# Output: 4.47213595499958

In the above code, we calculated the Euclidean distance between two points. 2 and 4.

Behind the scenes, it will calculate like this: sqrt(2^2 + 4^2) = sqrt(4 + 16) = sqrt(20) and square root of 20 is 4.4721.

So, concepts like Euclidean distance (L2 norm) or Manhattan distance (L1 norm) are vector norms.

Syntax

numpy.linalg.norm(x, ord=None, axis=None, keepdims=False)

Parameters

Argument	Description
x (array_like, required)	It represents an array or scalar. It must be 1D or 2D.
ord (optional, default=None)	It specifies the norm order. Vector norms: ord=None: 2-norm (Euclidean norm). ord=1: 1-norm (sum of absolute values). ord=2: 2-norm (Euclidean). ord=∞: max(abs(x)). ord=-∞: min(abs(x)). Matrix norms: ord=None: Frobenius norm. ‘fro’: Frobenius norm. ‘nuc’: nuclear norm (sum of singular values). ord=1: max column sum. ord=∞: max row sum. ord=2: 2-norm (largest singular value).
axis (optional, default=None)	It represents an axis or axes over which to compute the norm.
keepdims (bool, optional, default=False)	If it is True, it retains the reduced axes in the output as singleton dimensions.

1-norm (Manhattan), Infinity norm, General p-norm (p=3)

Let’s calculate the Manhattan norm, the infinity norm, and the general p-norm for vectors.

import numpy as np

vector = np.array([3, 4])

# 1-norm (Manhattan)
manhattan_norm = np.linalg.norm(vector, ord=1)
print(manhattan_norm)
# Output: 7.0 (|3| + |4| = 7)

# Infinity norm (max absolute value)
infinity_norm = np.linalg.norm(vector, ord=np.inf)
print(infinity_norm)
# Output: 4.0 (max(|3|, |4|) = 4)

# General p-norm (p=3)
pnorm = np.linalg.norm(vector, ord=3)
print(pnorm)
# Output: 4.497941445275415
# ∣1∣^3=1,∣2∣^3=8,∣3∣^3 = 27
# 1+8+27 = 36 (Sum)
# 36^(1/3) = 4.497941445275415 (Cube Root)

You can see that for different types of norms, we passed different ord values. For example, for 1-norm, we passed ord=1, for infinity norm, we passed ord=np.inf, and for p-norm, we passed ord=3.

Matrix norms

Let’s find the Frobenius (default for matrices) norm of the flattened matrix.

import numpy as np

mat = np.array([[1, 2], [3, 4]])

matrix_norm = np.linalg.norm(mat)

print(matrix_norm)

# Output: 5.477225575051661

Here is the explanation for the output: sqrt(1^2 + 2^2 + 3^2 + 4^2) = 5.47722

1-norm (max column sum), Infinity norm (max row sum), and Nuclear norm

1-norm is the maximum absolute column sum of the matrix.

∞-norm is the maximum absolute row sum of the matrix.

The nuclear norm is the sum of singular values.

import numpy as np 

mat = np.array([[1, 2], [3, 4]])

# 1-norm (max column sum)
print(np.linalg.norm(mat, ord=1))   
# Output: max(|1|+|3|, |2|+|4|) = 6.0

# Infinity norm (max row sum)
print(np.linalg.norm(mat, ord=np.inf))  
# Output: max(|1|+|2|, |3|+|4|) = 7.0

# Nuclear norm (sum of singular values)
print(np.linalg.norm(mat, ord='nuc'))   
# Output: 5.8309518948453

By passing ord=1, we calculated the 1-norm of a matrix.

By passing ord=np.inf, we calculated the infinity norm of an input matrix.

And finally, for nuclear norm, we passed ord=nuc.

That’s all!