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Concept
Singular value decomposition
Summary
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition.
Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V. Such decomposition always exists for any complex matrix. If M is real, then U and V can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted
The diagonal entries of are uniquely determined by M and are known as the singular values of M. The number of non-zero singular values is equal to the rank of M. The columns of U and the columns of V are called left-singular vectors and right-singular vectors of M, respectively. They form two sets of orthonormal bases u_1, ..., u_m and v_1, ..., v_n , and if they are sorted so that the singular values with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as where is the rank of M.
The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not U and V) is uniquely determined by M.
The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the non-zero singular values. In this variant, U is an semi-unitary matrix and is an semi-unitary matrix, such that
Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.
Official source
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