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