# Singular value decomposition

Formally, the singular value decomposition of an ${\displaystyle m\times n}$ complex matrix M is a factorization of the form ${\displaystyle \mathbf {U\Sigma V^{*}} }$, where U is an ${\displaystyle m\times m}$ complex unitary matrix, ${\displaystyle \mathbf {\Sigma } }$ is an ${\displaystyle m\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an ${\displaystyle n\times n}$ complex unitary matrix. The diagonal entries ${\displaystyle \sigma _{i}=\Sigma _{ii}}$ of ${\displaystyle \mathbf {\Sigma } }$ are known as the singular values of M. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of M, respectively.