[Math Review] Linear Algebra for Singular Value Decomposition (SVD)

Matrix and Determinant

Let C be an M × N matrix with real-valued entries, i.e. C={c_ij}_mxn

Determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or |A|.

In the case of a 2 × 2 matrix the determinant may be defined as:

Similarly, for a 3 × 3 matrix A, its determinant is:

See more information about determinant here.

Rank of Matrix

The Rank of a matrix is the number of linearly independent rows (or columns) in it, so rank(C)≤min(m,n).

A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally row echelon form, by elementary row operations. The rank equals to the number of non-zero rows of the final matrix (in row echelon form).

The reduce step can be found in this article.

Eigenvalues and Eigenvectors

For a square M × M matrix C and a vector x that is not all zeros, the values of λ satisfying 

 

are called the eigenvalues of C . The N-vector ⃗x satisfying the equation above for an eigenvalue λ is the corresponding right eigenvector.

How to Calculate

The eigenvalues of C are then the solutions of

|(C − λI_M)| = 0,

where |S| denotes the determinant of a square matrix S. 

For each value of  λ, we can calculate the corresponding eigenvector x through solving the following equation:

This article gives a specific example of the calculating process.

Matrix Decompositions

Matrix diagonalization theorem

Let S be a square real-valued M × M matrix with M linearly independent eigenvectors. Then there exists an eigen decomposition

where the columns of U are the eigenvectors of S and Λ is a diagonal matrix whose diagonal entries are the eigenvalues of S in decreasing order

If the eigenvalues are distinct, then this decomposition is unique.

Symmetric diagonalization theorem

Let S be a square, symmetric real-valued M × M matrix with M linearly independent eigenvectors. Then there exists a symmetric diagonal decomposition 

S = QΛQ^T

where the columns of Q are the orthogonal and normalized (unit length, real) eigenvectors of S, and Λ is the diagonal matrix whose entries are the eigenvalues of S.

Further, all entries of Q are real and we have Q⁻¹ = Q^T. 

Singular value decompositions 

Let r be the rank of the M × N matrix C. Then, there is a singular- value decomposition (SVD for short) of C of the form

where

1. U is the M × M matrix whose columns are the orthogonal eigenvectors of CC^T.

2. V is the N × N matrix whose columns are the orthogonal eigenvectors of C^TC. 

3. 

The values σ_i are referred to as the singular values of C. 

 Here is the illustration of the singular-value decomposition.