Singular Value Decomposition

Singular Value Decomposition is one of the important concepts in linear algebra. To understand the meaning of singular value decomposition (SVD), one must be aware of the related concepts such as matrix, types of matrices, transformations of a matrix, etc. As this concept is connected to various concepts of linear algebra, it’s become challenging to learn the singular value decomposition of a matrix. In this article, you will learn the definition of singular value decomposition, examples of 2×2 and 3×3 matrix decomposition in detail.

What is Singular Value Decomposition?

The Singular Value Decomposition of a matrix is a factorization of the matrix into three matrices. Thus, the singular value decomposition of matrix A can be expressed in terms of the factorization of A into the product of three matrices as A = UDV^T

Here, the columns of U and V are orthonormal, and the matrix D is diagonal with real positive entries.

Singular Value Decomposition of a Matrix

Mathematically, the singular value decomposition of a matrix can be explained as follows:

Consider a matrix A of order mxn.

This can be uniquely decomposed as:

A = UDV^T

U is mxn and column orthogonal (that means its columns are eigenvectors of AA^T)

(AA^T = UDV^T VDU^T = UD²U^T )

V is nxn and orthogonal (that means its columns are eigenvectors of A^TA)

(A^TA = VDU^T UDV^T = VD²V ^T )

D is nxn diagonal, where non-negative real values are called singular values.

Learn how to find eigenvalues and eigenvectors of a matrix here.

Let D = diag(σ₁, σ₂,…, σ_n) ordered such that σ₁ ≥ σ₂ ≥ … ≥ σ_n.

If σ is a singular value of A, its square is an eigenvalue of A^TA.

Also, let U = (u₁ u₂ … u_n) and V = (v₁ v₂ … v_n).

Therefore,

Here, the sum can be given from 1 to r so that r is the rank of matrix A.

Go through the example given below to understand the process of singular value decomposition of a matrix in a better way.

Singular Value Decomposition Example

The process of finding the singular value decomposition for 3×3 matrix and 2×2 matrix is the same. Let’s have a look at the example of 2×2 matrix decomposition.

Singular Value Decomposition 2×2 Matrix Example

Question:

Find the singular value decomposition of a matrix

Solution:

Given,

So,

Now,

Finding the eigenvector for AAT.

∴ The eigenvalues of the matrix A⋅A′ are given by λ = 1, 81.

Now,

Eigenvectors for λ = 81 are:

Eigenvectors for λ = 1 are:

Similarly, we can find the eigenvectors for A’A as:

Eigenvectors for λ = 81 are:

Eigenvectors for λ = 1 are:

Using these values, we can write the solution.

Or

Singular Value Decomposition Applications

Some of the applications of singular value decomposition are listed below:

• SVD has some fascinating algebraic characteristics and conveys relevant geometrical and theoretical insights regarding linear transformations.
• SVD has some critical applications in data science too.
• Mathematical applications of the SVD involve calculating the matrix approximation, rank of a matrix and so on.
• The SVD is also greatly useful in science and engineering.
• It has some applications of statistics, for example, least-squares fitting of data and process control.

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