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 matrix via an extension of polar decomposition. It has many useful applications in signal processing and statistics. 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 a 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. The singular value decomposition can be computed using the following observations: \n* The left-singular vectors of M are a set of orthonormal eigenvectors of MM∗. \n* The right-singular vectors of M are a set of orthonormal eigenvectors of M∗M. \n* 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.

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