\[A_{m*n} = U_{m*m} \Sigma_{m*n} V_{n*n}^T\]

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any \({\displaystyle m\times n} m\times n\) matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.

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

The singular value decomposition can be computed using the following observations:

The left-singular vectors of M are a set of orthonormal eigenvectors of MM∗. The right-singular vectors of M are a set of orthonormal eigenvectors of M∗M. The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both M∗M and MM∗. Applications that employ the SVD include computing the pseudoinverse, least squares fitting of data, multivariable control, Matrix Approximation, and determining the rank, range and null space of a matrix.