text: It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any m n {\displaystyle \ m\times n\ } matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m n {\displaystyle \ m\times n\ } complex matrix M is a factorization of the form M = U V , {\displaystyle \ \mathbf {M} =\mathbf {U\Sigma V^{*}} \ ,} where U is an m m {\displaystyle \ m\times m\ } complex unitary matrix, {\displaystyle \ \mathbf {\Sigma } \ } is an m n {\displaystyle \ m\times n\ } rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an n n {\displaystyle n\times n} complex unitary matrix, and V {\displaystyle \ \mathbf {V^{*}} \ } is the conjugate transpose of V. Such decomposition always exists for any complex matrix. If M is real, then U and V can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted U V T {\displaystyle \ \mathbf {U\Sigma V} ^{\operatorname {T} }} . The diagonal entries i = i i {\displaystyle \ \sigma _{i}=\Sigma _{ii}\ } of {\displaystyle \ \mathbf {\Sigma } \ } are uniquely determined by M and are known as the singular values of M. The number of non-zero singular values is equal to the rank of M. The columns of U and the columns of V are called left-singular vectors and right-singular vectors of M, respectively. They form two sets of orthonormal bases u1, ..., um and v1, ..., vn , and if they are sorted so that the singular values i {\displaystyle \ \sigma _{i}\ } with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as M = i = 1 r i u i v i , {\displaystyle \ \mathbf {M} =\sum _{i=1}^{r}\sigma _{i}\mathbf {u} _{i}\mathbf {v} _{i}^{*}\ ,} where r min { m , n } {\displaystyle \ r\leq \min\{m,n\}\ } is the rank of M. The SVD is not unique. It is always possible to choose the decomposition so that the singular values i i {\displaystyle \Sigma _{ii}} are in descending order. In this case, {\displaystyle \mathbf {\Sigma } } (but not U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition M = U V {\displaystyle \ \mathbf {M} =\mathbf {U\Sigma V^{*}} \ } in which {\displaystyle \ \mathbf {\Sigma } \ } is square diagonal of size r r {\displaystyle r\times r} , where r min { m , n } {\displaystyle \ r\leq \min\{m,n\}\ } is the rank of M, and has only the non-zero singular values. In this variant, U is an m r {\displaystyle m\times r} semi-unitary matrix and V {\displaystyle \ \mathbf {V} \ } is an n r {\displaystyle n\times r} semi-unitary matrix, such that U U = V V = I r . {\displaystyle \ \mathbf {U^{*}U} =\mathbf {V^{*}V} =\mathbf {I} _{r}\ .} Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.