Singular Values
Singular values are a generalization of eigenvalues.
Contents
Introduction
It is known that (for square matrices) there are certain vectors which the transformation only scales: Ax = λx.
Similarly, for any matrix A of size m x n, there are certain vectors ui in the column space that correspond to certain vectors vi in the row space. They correspond such that Av = σu. And by composing these vectors and values together: AV = UΣ.
Description
Singular values are square roots of eigenvalues. Specifically, a matrix A has singular values equal to the square roots of the eigenvalues of ATA: σ = √λ.
Matrices of all sizes have singular values. The number of non-zero singular values is equal to the rank of A.