Formally, the singular-value decomposition of an  real or complex matrix  is a factorization of the form , where  is an  real or complex unitary matrix,  is an  rectangular diagonal matrix with non-negative real numbers on the diagonal, and  is an  real or complex unitary matrix. The diagonal entries  of  are known as the singular values of . The columns of  and the columns of  are called the left-singular vectors and right-singular vectors of , respectively.