Singular Values#

Theorem: Singular Value Decomposition

Every real matrix \(A \in \mathbb{R}^{m \times n}\) can be represented as the product of a diagonal matrix \(\Sigma \in \mathbb{R}^{m \times n}\) with two orthogonal matrices \(U \in \mathbb{R}^{m \times m}\) and \(V \in \mathbb{R}^{n\times n}\) as follows:

Moreover, the entries on \(\Sigma\)'s diagonal are \(A\)'s (not necessarily distinct) singular values.