What is "singular value decomposition"

n. (context linear algebra English) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entry on its diagonal the singular values of the original matrix.

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 m × n matrix via an extension of polar decomposition. It has many useful applications in signal processing and statistics.

Formally, the singular value decomposition of an m × n real or complex matrix M is a factorization of the form U\Sigma V^*, where U is an m × m real or complex unitary matrix, \Sigma is a m × n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n × n real or complex unitary matrix. The diagonal entries σ of \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.

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

• The left-singular vectors of are a set of orthonormal eigenvectors of .
• The right-singular vectors of are a set of orthonormal eigenvectors of .
• The non-zero singular values of (found on the diagonal entries of ) are the square roots of the non-zero eigenvalues of both and .

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.