Compute the condition number of a matrix in linear algebra in Python

The condition number of a matrix measures how sensitive the solution of a linear system is to changes in the input. A low condition number indicates a well-conditioned matrix, while a high condition number suggests an ill-conditioned matrix. In Python, we use numpy.linalg.cond() to compute this value.

Syntax

numpy.linalg.cond(x, p=None)

Parameters

x: The matrix whose condition number is sought.
p: Order of the norm used in computation (None, 1, -1, 2, -2, 'fro').

Basic Example

Let's compute the condition number of a 3x3 matrix ?

import numpy as np
from numpy import linalg as LA

# Create a matrix
matrix = np.array([[1, 1, 0],
                   [1, 0, 1],
                   [1, 0, 0]])

print("Matrix:")
print(matrix)

# Compute condition number
cond_num = LA.cond(matrix)
print(f"\nCondition Number: {cond_num}")

Matrix:
[[1 1 0]
 [1 0 1]
 [1 0 0]]

Condition Number: 3.7320508075688776

Using Different Norms

The condition number can be calculated using different norms ?

import numpy as np
from numpy import linalg as LA

matrix = np.array([[2, 1],
                   [1, 2]])

print("Matrix:")
print(matrix)

# Different norm types
norms = [None, 1, -1, 2, 'fro']
for norm in norms:
    cond_num = LA.cond(matrix, p=norm)
    print(f"Condition number (p={norm}): {cond_num:.4f}")

Matrix:
[[2 1]
 [1 2]]

Condition number (p=None): 3.0000
Condition number (p=1): 3.0000
Condition number (p=-1): 3.0000
Condition number (p=2): 3.0000
Condition number (p=fro): 3.0000

Well-conditioned vs Ill-conditioned Matrices

Let's compare a well-conditioned matrix with an ill-conditioned one ?

import numpy as np
from numpy import linalg as LA

# Well-conditioned matrix (identity matrix)
well_conditioned = np.eye(3)
print("Well-conditioned matrix:")
print(well_conditioned)
print(f"Condition number: {LA.cond(well_conditioned):.4f}")

# Ill-conditioned matrix
ill_conditioned = np.array([[1, 1, 1],
                           [1, 1.0001, 1],
                           [1, 1, 1.0001]])
print("\nIll-conditioned matrix:")
print(ill_conditioned)
print(f"Condition number: {LA.cond(ill_conditioned):.4f}")

Well-conditioned matrix:
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
Condition number: 1.0000

Ill-conditioned matrix:
[[1.      1.      1.     ]
 [1.      1.0001  1.     ]
 [1.      1.      1.0001 ]]
Condition number: 40001.0000

Key Points

Conclusion

The condition number is crucial for assessing numerical stability in linear algebra computations. Use numpy.linalg.cond() to evaluate matrix conditioning before solving linear systems or performing matrix inversions.