Singular Value Decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. 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 (the conjugate transpose of , or simply the transpose of if is real) 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 and the eigendecomposition are closely related. Namely: * The left-singular vectors of are eigenvectors of . * The right-singular vectors of are 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.