Return the Norm of the vector over given axis in Linear Algebra in Python

The norm of a vector or matrix measures its magnitude or size. In NumPy, you can calculate various types of norms using numpy.linalg.norm(), including vector norms along specific axes.

Syntax

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

Parameters

x: Input array (vector or matrix)
ord: Order of the norm (default is 2-norm)
axis: Axis along which to compute the norm
keepdims: Whether to keep dimensions in the result

Basic Vector Norm Example

Let's start with calculating the norm of a simple vector ?

import numpy as np
from numpy import linalg as LA

# Create a simple vector
vector = np.array([3, 4, 5])
print("Vector:", vector)

# Calculate 2-norm (default)
norm_2 = LA.norm(vector)
print("2-norm:", norm_2)

# Calculate 1-norm (Manhattan distance)
norm_1 = LA.norm(vector, ord=1)
print("1-norm:", norm_1)

# Calculate infinity norm (maximum absolute value)
norm_inf = LA.norm(vector, ord=np.inf)
print("Infinity norm:", norm_inf)

Vector: [3 4 5]
2-norm: 7.0710678118654755
1-norm: 12.0
Infinity norm: 5.0

Matrix Norm Along Specific Axis

For matrices, you can compute norms along specific axes ?

import numpy as np
from numpy import linalg as LA

# Create a matrix
matrix = np.array([[-4, -3, -2],
                   [-1,  0,  1],
                   [ 2,  3,  4]])

print("Matrix:")
print(matrix)

# Norm along axis=0 (column-wise)
norm_axis0 = LA.norm(matrix, axis=0)
print("\nNorm along axis=0 (columns):", norm_axis0)

# Norm along axis=1 (row-wise)
norm_axis1 = LA.norm(matrix, axis=1)
print("Norm along axis=1 (rows):", norm_axis1)

# Overall matrix norm (Frobenius norm)
frobenius_norm = LA.norm(matrix)
print("Frobenius norm:", frobenius_norm)

Matrix:
[[-4 -3 -2]
 [-1  0  1]
 [ 2  3  4]]

Norm along axis=0 (columns): [4.58257569 4.24264069 4.58257569]
Norm along axis=1 (rows): [5.38516481 1.41421356 5.38516481]
Frobenius norm: 9.273618495495704

Using Different Norm Orders

Different ord values give different types of norms ?

import numpy as np
from numpy import linalg as LA

matrix = np.array([[-4, -3, -2],
                   [-1,  0,  1],
                   [ 2,  3,  4]])

# Different matrix norms
print("Matrix infinity norm:", LA.norm(matrix, ord=np.inf))
print("Matrix 1-norm:", LA.norm(matrix, ord=1))
print("Matrix 2-norm:", LA.norm(matrix, ord=2))
print("Matrix -1 norm:", LA.norm(matrix, ord=-1))

Matrix infinity norm: 9.0
Matrix 1-norm: 7.0
Matrix 2-norm: 7.348469228349534
Matrix -1 norm: 4.0

Common Norm Types

Conclusion

Use numpy.linalg.norm() to calculate vector and matrix norms. Specify axis parameter to compute norms along specific dimensions, and use different ord values for various norm types like L1, L2, and infinity norms.