Singular Value Decomposition#
Definition: Singular Values
Let \(A \in \mathbb{C}^{m \times n}\) be a complex matrix.
The singular values of \(A\) are the square roots of the eigenvalues of the matrix product \(A^{\ast} A\) between \(A\) and its Hermitian transpose.
Theorem: Singular Value Decomposition
Every complex matrix \(A \in \mathbb{C}^{m \times n}\) can be expressed as the product of a unitary matrix \(U \in \mathbb{C}^{m \times m}\), a diagonal matrix \(\Sigma \in \mathbb{C}^{m \times n}\) and the Hermitian transpose of a unitary matrix \(V \in \mathbb{C}^{n \times n}\):
Moreover, the entries on \(\Sigma\)'s diagonal are precisely the (not necessarily distinct) singular values of \(A\). If \(\sigma_{j} = \sqrt{\lambda_j}\) is the diagonal entry on the \(j\)-th column of \(\Sigma\), then the \(j\)-th column of \(V\) is an eigenvector of \(A^{\ast}A\) which corresponds to \(\lambda_j\).
Proof
TODO