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- Singular_value_decomposition abstract "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 m × n real or complex matrix M is a factorization of the form M = UΣV∗, where U is an m × m real or complex unitary matrix, Σ is an m × n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V∗ (the conjugate transpose of V, or simply the transpose of V if V is real) is an n × n real or complex unitary matrix. The diagonal entries Σi,i of Σ are known as the singular values of M. The m columns of U and the n columns of V are called the left-singular vectors and right-singular vectors of M, respectively.The singular value decomposition and the eigendecomposition are closely related. Namely: The left-singular vectors of M are eigenvectors of MM∗. The right-singular vectors of M are 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.".
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