SciPy - Frobenius Norm(fro)

---

---

The Frobenius Norm in SciPy is a specific type of matrix norm widely used in numerical linear algebra. It measures the size or magnitude of a matrix by summing the squared magnitudes of all its elements. It is especially useful in problems involving error estimation, optimization and linear algebra.

The Frobenius norm is equivalent to the Euclidean norm of the matrix when the matrix is treated as a vector by stacking its rows or columns.

When given matrix is A with dimensions m x n, then the Mathematical formula for the Frobenius Norm ||A||_F is defined as follows −

‖A‖F = √(∑i=1m ∑j=1n |aij|2)

Where −

• a_ij represents the element in the i-th row and j-th column of A.
• The norm computes the sum of the squared absolute values of all matrix elements followed by taking the square root.

Key Properties of the Frobenius Norm

The Frobenius norm has several important properties that make it useful in mathematics, numerical linear algebra and applications involving matrix computations. Following are the key Properties of the Frobenius Norm −

Non-Negativity

The Frobenius norm is always non-negative and the equality holds if and only if A is a zero matrix (A_ij = 0 for all i, j).

AF  0

Homogeneity (Scaling)

The Frobenius norm is homogeneous with respect to scalar multiplication.

cAF = |c| . AF

where c is a scalar and A is a matrix. This means that scaling a matrix by a factor c scales its Frobenius norm by c.

Subadditivity (Triangle Inequality)

The Frobenius norm satisfies the triangle inequality.

A + BF  |A|F + BF

where A and B are matrices of the same size. This property ensures that the Frobenius norm behaves consistently with distance metrics.

Unitary/Orthogonal Invariance

The Frobenius norm is invariant under multiplication by orthogonal (real matrices) or unitary (complex matrices) matrices.

UAF = |A|F and AVF = |A|F

where U and V are orthogonal/unitary matrices. This property is crucial for numerical methods like Singular Value Decomposition (SVD) as it ensures the norm remains unchanged under certain transformations.

Relationship with Vector Norm

If a matrix A is reshaped into a vector vec(A) then the Frobenius norm of A equals the Euclidean (L2) norm of vec(A) −

AF = vec(A)2

This property highlights the connection between matrix norms and vector norms.

Example 1

Following is the example of the Frobenius Norm calculated with the help of the function scipy.linalg.norm() by passing the parameter ord = 'fro'. Here in this example we are calculating the Frobenius Norm for a small 2d matrix −

import numpy as np
from scipy.linalg import norm

# Define the matrix
A = np.array([[1, 2],
              [3, 4]])

# Compute the Frobenius norm
fro_norm = norm(A, ord='fro')

print("Matrix A:")
print(A)
print("Frobenius Norm of A:", fro_norm)

Following is the output of the Frobenius Norm calculated for a 2D matrix −

Matrix A:
[[1 2]
 [3 4]]
Frobenius Norm of A: 5.477225575051661

Example 2

Here in this example we are going to calculate the Frobenius Norm for a complex matrix with the use of scipy.linalg.norm() function −

import numpy as np
from scipy.linalg import norm

# Define the complex matrix
A = np.array([[1 + 1j, 2 - 1j],
              [-1j, 3 + 2j]])

# Compute the Frobenius norm
fro_norm = norm(A, ord='fro')

print("Complex Matrix A:")
print(A)
print("Frobenius Norm of A:", fro_norm)

Below is the output of the Frobenius Norm calculated for a complex matrix −

Complex Matrix A:
[[ 1.+1.j  2.-1.j]
 [-0.-1.j  3.+2.j]]
Frobenius Norm of A: 4.58257569495584

Example 3

Heres an example of calculating the Frobenius norm for a zero matrix −

import numpy as np
from scipy.linalg import norm

# Define the zero matrix
A = np.zeros((2, 2))

# Compute the Frobenius norm
fro_norm = norm(A, ord='fro')

print("Zero Matrix A:")
print(A)
print("Frobenius Norm of A:", fro_norm)

Here is the output of the Frobenius Norm calculated for a Zero matrix −

Zero Matrix A:
[[0. 0.]
 [0. 0.]]
Frobenius Norm of A: 0.0