An introduction to SVD

SVD utilizes the properties of eigenvectors and eigenvalues. SVD is a matrix decomposition method that reduces a large, usually sparse, matrix into three sub-matrices. We will assume the following:

• λ 1, λ 2, ..., λ n is the eigenvalues of a matrix A
• x 1, x 2, ..., x n is a set of corresponding eigenvectors in vector V
• Σ denotes the n x n diagonal matrix with the λ j on the diagonal
• X denotes the n x n matrix whose jth column is x j

Then, we can rewrite Eq. (1) as follows:

𝔸V = UΣ Eq. (2)

Eq. (2) is the matrix form of Eq. (1). In Eq. (2), it is necessary to put Σ as the second term on the right-hand side. This will make sure each column of X is multiplied by its corresponding eigenvalue. Let’s use a simple example to understand Eq. (2):

𝔸 = [4 0 0 0 3 0 0 0 2]

This matrix...