Singular value decomposition
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 matrixwith positive eigenvalues) to any 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 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.
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∗.