Singular Value Decomposition (SVD) using Python

by | Oct 3, 2023 | Featured, Linear Algebra

What is Singular Value Decomposition (SVD)?

Let A be an mxn rectangular matrix. Using Singular Value Decomposition (SVD), we can decompose the matrix A in the following way:

A_{m \times n}=U_{m \times m}S_{m \times n}V_{n \times n}^T

Here, U is an mxm matrix. S is an mxn matrix and VT is an nxn matrix.

To calculate the U, S, and VT matrices, we need to find out the eigen values and eigen vectors of AAT and ATA. The columns of the matrix U are the eigenvectors of AAT. And the columns of the matrix V are the eigenvectors of ATA. And the diagonal entries of S are the square roots of the eigenvalues of AAT and ATA. These diagonal entries of S are also called the singular values. And, if A is a real matrix, then U, S, and V matrices are also real.

Singular Value Decomposition (SVD) using Python

We can use the following Python code to perform Singular Value Decomposition (SVD) using Python.

import numpy

A = numpy.array([[1, 2], [2, 1], [0, 1]])

U, s, V_transpose = numpy.linalg.svd(A)
S = numpy.zeros((3, 2), numpy.float64)


numpy.fill_diagonal(S, s)
print("Matrix U: \n", U)
print("Matrix S: \n", S)
print("Matrix V_transpose: \n", V_transpose)

A_calculated = U.dot(S).dot(V_transpose)
if numpy.allclose(A_calculated, A):
    print("SVD is successful")
else:
    print("SVD is not successful")

Here, A is a 3×2 matrix. We are using numpy.linalg.svd() function to perform Singular Value Decomposition (SVD) of A. The function returns the matrices U and VT and the diagonal elements of S.

As S can be a rectangular matrix, we are first creating a null matrix and then, filling the diagonal of the null matrix with the returned diagonal elements s.

We are then calculating the matrix USVT and using the function numpy.allclose() to check the calculated matrix is the same as A element-wise.

The output of the given program will be:

Matrix U: 
 [[-0.70002658  0.47354883 -0.53452248]
 [-0.67156256 -0.69106812  0.26726124]
 [-0.2428302   0.54605526  0.80178373]]
Matrix S: 
 [[3.08725264 0.        ]
 [0.         1.21196994]
 [0.         0.        ]]
Matrix V_transpose: 
 [[-0.66180256 -0.74967818]
 [-0.74967818  0.66180256]]
SVD is successful
Facebooktwitterredditpinterestlinkedinmail

Calculate the pseudoinverse of a matrix using Python

What is the pseudoinverse of a matrix? We know that if A is a square matrix with full rank, then A-1 is said to be the inverse of A if the following condition holds: $latex AA^{-1}=A^{-1}A=I $ The pseudoinverse or the Moore-Penrose inverse of a matrix is a...

Cholesky decomposition using Python

What is Cholesky decomposition? A square matrix A is said to have Cholesky decomposition if it can be written as a product of a lower triangular matrix and its conjugate transpose. $latex A=LL^{*} $ If all the entries of A are real numbers, then the conjugate...

Tensor Hadamard Product using Python

In one of our previous articles, we already discussed what the Hadamard product in linear algebra is. We discussed that if A and B are two matrices of size mxn, then the Hadamard product of A and B is another mxn matrix C such that: $latex H_{i,j}=A_{i,j} \times...

Perform tensor addition and subtraction using Python

We can use numpy nd-array to create a tensor in Python. We can use the following Python code to perform tensor addition and subtraction. import numpy A = numpy.random.randint(low=1, high=10, size=(3, 3, 3)) B = numpy.random.randint(low=1, high=10, size=(3, 3, 3)) C =...

How to create a tensor using Python?

What is a tensor? A tensor is a generalization of vectors and matrices. It is easily understood as a multidimensional array. For example, in machine learning, we can organize data in an m-way array and refer it as a data tensor. Data related to images, sounds, movies,...

How to combine NumPy arrays using horizontal stack?

We can use the hstack() function from the numpy module to combine two or more NumPy arrays horizontally. For example, we can use the following Python code to combine three NumPy arrays horizontally. import numpy A = numpy.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) B =...

How to combine NumPy arrays using vertical stack?

Let’s say we have two or more NumPy arrays. We can combine these NumPy arrays vertically using the vstack() function from the numpy module. For example, we can use the following Python code to combine three NumPy arrays vertically. import numpy A = numpy.array([[1, 2,...

Eigen decomposition of a square matrix using Python

Let A be a square matrix. Let’s say A has k eigenvalues λ1, λ2, ... λk. And the corresponding eigenvectors are X1, X2, ... Xk. $latex X_1=\begin{bmatrix} x_{11} \\ x_{21} \\ x_{31} \\ ... \\ x_{k1} \end{bmatrix} \\ X_2=\begin{bmatrix} x_{12} \\ x_{22} \\ x_{32} \\ ......

How to calculate eigenvalues and eigenvectors using Python?

In our previous article, we discussed what eigen values and eigenvectors of a square matrix are and how we can calculate the eigenvalues and eigenvectors of a square matrix mathematically. We discussed that if A is a square matrix, then $latex (A- \lambda I) \vec{u}=0...

What are eigenvalues and eigenvectors?

What are eigenvalues and eigenvectors? Let’s say A is a square matrix. Now, if we multiply A with a vector u, we will get another vector v. $latex A \vec{u}= \vec{v} $ The direction of the vectors u and v will be different in most cases. In other words, if we multiply...

Amrita Mitra

Author

Ms. Amrita Mitra is an author, who has authored the books “Cryptography And Public Key Infrastructure“, “Web Application Vulnerabilities And Prevention“, “A Guide To Cyber Security” and “Phishing: Detection, Analysis And Prevention“. She is also the founder of Asigosec Technologies, the company that owns The Security Buddy.

0 Comments

Submit a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Not a premium member yet?

Please follow the link below to buy The Security Buddy Premium Membership.

Featured Posts

Translate »