The norm of a matrix | Pythontic.com

Overview:

The norm of a square matrix describes the magnitude of the matrix.
Matrix norm is a function that combines all the values of a matrix to form a single positive real number that satisfies the following conditions.
- || A || >= 0 for any square matrix A
- || A || >= 0 if and only if the matrix A = 0
- || kA || = | k | || A || where k is a scalar
- || A + B || <= || A || + || B ||
- || AB || <= || A || || B ||
There are several matrix norms and each of them are calculated differently.
The parameter ord of the numpy.linalg.norm() function denotes which matrix norm needs to be calculated.

Types of matrix norms:

Frobenius norm or Euclidean norm: Returns the square root of the sum of the squares of the

all matrix elements. This is the default norm returned by the numpy.norm() function when ord = None.
The 1-norm: The 1-norm is calculated by taking the maximum of the sums of the matrix columns. Column sums are calculated using the absolute value of the column elements.
The -1-norm: The -1-norm is calculated by taking the minimum of the sums of the matrix columns. Column sums are calculated using the absolute value of the column elements.
The 2-norm: Conventionally refers to the Euclidean norm. However when ord = 2, numpy.linalg.norm() returns the maximum of the singular values of the diagonal matrix obtained through singular value decomposition.
The -2-norm: Returns the minimum of the singular values of the diagonal matrix obtained through singular value decomposition.
Nuclear norm: The Nuclear norm is provided by summing up the singular values obtained through singular value decomposition.

Example:

# Example Python program that finds the various matrix norms of a given

# matrix using the function norm() of numpy.linalg module
import numpy
import math

# Create a 2x2 square matrix
a1 = numpy.ndarray(shape = (2, 2),
dtype = numpy.int32)

a1[0][0] = -2
a1[0][1] = -4
a1[1][0] = 3
a1[1][1] = 5

print("Square matrix:")
print(a1)

# Frobenius norm / Euclidean norm
normVal = numpy.linalg.norm(a1)
print("Euclidean norm of the matrix:")
print(normVal)

# When ord = 1
normVal = numpy.linalg.norm(a1, ord = 1)
print("1-norm of the matrix:")
print(normVal)

# When ord = -1
normVal = numpy.linalg.norm(a1, ord = -1)
print("-1-norm of the matrix:")
print(normVal)

# When ord = 2
normVal = numpy.linalg.norm(a1, ord = -2)
print("2-norm of the matrix:")
print(normVal)

# When ord = -2
normVal = numpy.linalg.norm(a1, ord = -2)
print("-2-norm of the matrix:")
print(normVal)

# When ord=inf
normVal = numpy.linalg.norm(a1, ord=numpy.inf)
print("Infinity norm of the matrix:")
print(normVal)

# When ord=-inf
normVal = numpy.linalg.norm(a1, ord=-numpy.inf)
print("Negative infinity norm of the matrix:")
print(normVal)

Output:

Square matrix:
[[-2 -4]
[ 3 5]]
Euclidean norm of the matrix:
7.3484692283495345
1-norm of the matrix:
9.0
-1-norm of the matrix:
5.0
2-norm of the matrix:
0.272352646999216
-2-norm of the matrix:
0.272352646999216
Infinity norm of the matrix:
8.0
Negative infinity norm of the matrix:
6.0