Return matrix rank of array using Singular Value Decomposition method in Python

To return the matrix rank of an array using the Singular Value Decomposition (SVD) method, use the numpy.linalg.matrix_rank() method in Python. The rank of a matrix represents the number of linearly independent rows or columns, calculated as the count of singular values greater than a specified tolerance.

Syntax

numpy.linalg.matrix_rank(A, tol=None, hermitian=False)

Parameters

A: Input vector or stack of matrices whose rank needs to be computed.

tol: Threshold below which SVD values are considered zero. If None, it's automatically set to S.max() * max(M, N) * eps, where S contains singular values and eps is the machine precision.

hermitian: If True, assumes the matrix is Hermitian (complex conjugate transpose equals itself), enabling more efficient computation. Defaults to False.

Example

Let's compute the matrix rank of an identity matrix using SVD ?

import numpy as np
from numpy.linalg import matrix_rank

# Create a 5x5 identity matrix
arr = np.eye(5)

# Display the array
print("Our Array...")
print(arr)

# Check dimensions and properties
print("\nDimensions:", arr.ndim)
print("Datatype:", arr.dtype)
print("Shape:", arr.shape)

# Calculate matrix rank using SVD
rank = matrix_rank(arr)
print("\nMatrix Rank:", rank)

Our Array...
[[1. 0. 0. 0. 0.]
 [0. 1. 0. 0. 0.]
 [0. 0. 1. 0. 0.]
 [0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 1.]]

Dimensions: 2
Datatype: float64
Shape: (5, 5)

Matrix Rank: 5

Working with Different Matrices

Let's see how matrix rank varies for different types of matrices ?

import numpy as np
from numpy.linalg import matrix_rank

# Create different matrices
identity = np.eye(3)
singular = np.array([[1, 2, 3], [2, 4, 6], [1, 2, 3]])  # Rank deficient
random_matrix = np.random.rand(3, 3)

print("Identity Matrix Rank:", matrix_rank(identity))
print("Singular Matrix Rank:", matrix_rank(singular))
print("Random Matrix Rank:", matrix_rank(random_matrix))

# Using custom tolerance
print("Singular Matrix Rank (tol=0.1):", matrix_rank(singular, tol=0.1))

Identity Matrix Rank: 3
Singular Matrix Rank: 1
Random Matrix Rank: 3
Singular Matrix Rank (tol=0.1): 1

Key Points

• Identity matrices have full rank equal to their dimensions
• Singular (rank-deficient) matrices have rank less than their dimensions
• The tolerance parameter affects rank calculation for matrices with small singular values
• SVD provides numerical stability for rank computation

Conclusion

The numpy.linalg.matrix_rank() function efficiently computes matrix rank using SVD. Identity matrices have full rank, while singular matrices are rank-deficient. Adjusting the tolerance parameter helps handle numerical precision issues.