Condition number of a matrix

Overview:

The condition number of a matrix is defined as the product of its norm and the norm of its inverse.
k(A) = ||A||.||A^-1||
On a system of linear equations, the condition number gives the bound to which the solution will vary to a change in input data.
A smaller condition number means, a change in the input data even if it is large will result in a small amount of change in the output.
A larger condition number means, a change in the input data even if it is small will result in a large amount of change in the output.
For a singular matrix the condition number is infinity(∞)

Computing the condition number of a matrix using NumPy:

The NumPy module linalg has the function cond() that computes the condition number of a given matrix.
The matrix norm to be used can be specified through the parameter p of the cond() function. When no value is specified it uses the smallest of the singular values calculated using SVD.

Example:

# Example Python program that computes the
# condition number of a matrix
import numpy

# Create a square matrix
elems = numpy.arange(1, 18, 2)
matrix = elems.reshape(3, 3)
print("Matrix:")
print(matrix)

# Compute the condition number - uses SVD as norm
conditionNumber = numpy.linalg.cond(matrix)
print("Condition number computed using smallest of singular values as norm:")
print(conditionNumber)

# Compute the condition number using Frobenius norm
conditionNumber = numpy.linalg.cond(matrix, 'fro')
print("Condition number computed using Frobenius norm:")
print(conditionNumber)

Output:

Matrix:
[[ 1 3 5]
[ 7 9 11]
[13 15 17]]
Condition number computed using smallest of singular values as norm:
2.3721000095398664e+16
Condition number computed using Frobenius norm:
7.0095746850180696e+16