Singular value decomposition using NumPy

Overview:

The function svd() from the linalg module of NumPy library calculates the singular values of an input array A and returns the results contained in an SVDResult instance.
The first ndarray in the SVDResult instance contains the left singular vector.
The second ndarray in the SVDResult instance contains the singular values of the given array A.
The third ndarray in the SVDResult instance contains the right singular vector.

Singular Value Decomposition:

Using singular value decomposition the matrix A of size mxn is factorized into
- An orthogonal matrix U of dimension mxm.
- A complex conjugate(also known as Hermitian transpose) of orthogonal matrix V of dimension nxn
- A diagonal matrix Σ of dimension mxn, containing singular values on its diagonal

as given by

A = UΣV^*

Unlike eigenvalues which are defined only for square matrices, singular values can be found for both square matrices and rectangular matrices.
The singular values are non-negative real numbers.
V^*- Complex conjugate of V.

Note:

Orthogonal matrix: An orthogonal matrix is a matrix when multiplied by its transpose results in an identity matrix.

Identity matrix: An identity matrix is a matrix whose diagonal elements are with the value one. An identity matrix is a diagonal matrix.

Diagonal matrix: A diagonal matrix is a matrix whose elements are all ones or zeroes on its diagonal.

Example:

# Example Python program that obtains the singular values of
# a square matrix
import numpy

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

a1[0][0] = 2
a1[0][1] = 5
a1[1][0] = -7
a1[1][1] = -3

# Obtain the singular values of the matrix
svdResults = numpy.linalg.svd(a1)
print(type(svdResults))
print(svdResults)

print("Left singular vector:")
print(numpy.linalg.svd(a1)[0])

print("Right singular vector:")
print(numpy.linalg.svd(a1)[2])

print("Singular values of the matrix:")
print(numpy.linalg.svd(a1)[1])

Output:

<class 'numpy.linalg._linalg.SVDResult'>
SVDResult(U=array([[-0.52573111, 0.85065081],
[ 0.85065081, 0.52573111]]), S=array([8.71337969, 3.32821489]), Vh=array([[-0.80405286, -0.59455781],
[-0.59455781, 0.80405286]]))
Left singular vector:
[[-0.52573111 0.85065081]
[ 0.85065081 0.52573111]]
Right singular vector:
[[-0.80405286 -0.59455781]
[-0.59455781 0.80405286]]
Singular values of the matrix:
[8.71337969 3.32821489]

Applications of Singular Value Decomposition:

Image processing
Text analysis