Singular Value DecompositionIt is an important matrix decomposition in linear algebra and has important applications in signal processing, statistics, and other fields. In some aspects, Singular Value Decomposition is similar to the symmetric matrix or the Hermite matrix based on feature vectors. However, although there is a correlation between the two matrix decomposition, there is a significant difference. The basis of feature vector decomposition of symmetric arrays is spectral analysis, while Singular Value Decomposition is the promotion of Spectral Analysis Theory on Arbitrary Matrices.
Directory
 1. theoretical description 2. Singular Values and singular vectors, and their relationship with Singular Value Decomposition
 1.1 intuitive explanation
 3. Relationship with feature Decomposition
 4. geometric meaning
 5. Simplified SVD
 6norm
 7 Applications
 7.1 pseudoInverse
 7.2 parallel Singular Value Model
 7.3 value range, zero space, and rank
 7.4 matrix Approximation
 8. Calculate SVD
 9. History
 10 See
 11 external links
 12 references

Theoretical description
HypothesisMIsM × nLevel matrix, where all elements belong to the domainK, That is, the real number field or the complex number field. So there is a decomposition
WhereUYesM × mOrder Matrix; Σ is a semiDefinite MatrixM × nDiagonal matrix;V *, That isVIsN × nOrder Matrix. Such decomposition is calledMOfSingular Value Decomposition. Element Σ on the Σ diagonal lineI,IThat isMOfSingular Value.
A common practice is to arrange Singular Values in large and small order. So that Σ canMUniquely identified. (AlthoughUAndVStill uncertain .)
Intuitive explanation
In the matrixMIn the Singular Value Decomposition
 VColumns form a set of base vectors for orthogonal "input" or "analysis. These vectors are feature vectors.
 UColumns form a set of base vectors for orthogonal "output. These vectors are feature vectors.
 The element on the diagonal of Σ is a singular value, which can be regarded as a scalar "Expansion control" between input and output ". These are the characteristics of the sum and correspondUAndVCorresponding to the row vector.
Singular Values and singular vectors, and the relationship between them and Singular Value Decomposition and the link geometric meaning of feature value decomposition
BecauseUAndVVectors are all units vectors. We know thatUColumn vectorU1 ,...,UmConstituteKMA set of standard orthogonal basis of space. Similarly,VColumn vectorV1 ,...,VnIt also makes upKNA set of standard orthogonal basis of space (according to the standard dot product law of vector space ).
Linear transformationT:KN→KM, Put the VectorXConvertMx. Considering these standard orthogonal bases, this transformation is easy to describe:T(Vi) =σ I ui,I= 1,..., min (M,N), Whereσ IIs the numberIElements; whenI> Min (M,N,T(VI) = 0.
In this way, the geometric meaning of SVD theory can be summarized as follows: For each linear ingT:KN→KM,TSetKNTheIBase vector ing isKMTheIAnd map the remaining base vectors to zero vectors. Map these base VectorsTIt can be expressed as a nonnegative diagonal array.
Simplified SVD norm
1. The concept of matrix norm is set to A ε Cm × n, and A realvalue function is defined.  A  if the following conditions are met:
(1) Nonnegative:  A  ≥0, and  A  = 0 When and only when A = 0; (2) homogeneous:  aA  =  a  A , a, C; (3) Triangle Inequality:  A + B  ≤  A  +  B , A, bε Cm × n; (4) compatibility:  AB  ≤  A  B 
 A  is the matrix norm of. For example, if A = (aij) ε Cn × n is set
All are matrix norm.
Theorem 2: The matrix norms induced by the 1norm, 2norm, and ∞norm of the vector are
It is usually called column and norm, spectral norm, row, and norm in sequence.
Theorem 3: both the spectral norm and the Fnorm are undo norm, that is, for any matrix P and Q, yes  PAQ = A .
Application pseudoInverse
Singular Value Decomposition can be used to calculate the pseudoinverse of a matrix. If MatrixMIntoM=UΣV*, ThenMThe pseudoinverse is
In this example, Σ + transposes Σ and calculates the reciprocal of each nonzero element on its main diagonal. The pseudoinverse is usually used to solve the Linear Least Square problem.
Parallel Singular Value Model
The frequency selective fading channels are decomposed.
Approximate values of value range, zero space, and rank matrix
The main application of Singular Value Decomposition in statistics is Principal Component Analysis (PCA). It is a data analysis method used to find the hidden "pattern" in a large amount of data and can be used in pattern recognition, data compression. The PCA algorithm maps datasets to lowdimensional spaces. The feature values of a dataset (characterized by Singular Values in SVD) are arranged by importance. The Dimensionality Reduction Process is to discard unimportant feature vectors, the remaining feature vector space is the space after dimensionality reduction.
Calculate SVD
Matlab: [B c d] = svd (A) OpenCV: void cvSVD (CvArr * A, CvArr * W, CvArr * U = NULL, CvArr * V = NULL, int flags = 0)
For more information, see external links.
 LAPACK users manual gives details of subroutines to calculate the SVD (see also [1]).
 Applications of SVDon PC Hansen's web site.
 Introduction to the Singular Value Decompositionby Todd Will of the University of Wisconsin  La Crosse.
 Los Alamos group's book chapterhas helpful gene data analysis examples.
 MIT Lectureseries by Gilbert Strang. See Lecture #29 on the SVD.
 Java SVDlibrary routine.
 Java appletdemonstrating the SVD.
 Java scriptdemonstrating the SVD more extensively, paste your data from a spreadsheet.
 Chapter from "Numerical Recipes in C" gives more information about implementation and applications of SVD.
 Online Matrix Calculator Performs singular value decomposition of matrices.
References
 Demmel, J. and Kahan, W. (1990). Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy.Siam j. Sci. Statist. Comput.,11(5), 873912.
 Golub, G. h. and Van Loan, C. f. (1996 ). "Matrix Computations ". 3rd ed ., johns Hopkins University Press, Baltimore. ISBN 0801854148.
 Halldor, Bjornsson and Venegas, Silvia. (1997 ). "A manual for EOF and SVD analyses of climate data ". mcGill University, CCGCR Report No. 971, Montr éal, qué bec, 52pp.
 Hansen, P. C. (1987). The truncated SVD as a method for regularization.BIT,27, 534553.
 Horn, Roger A. and Johnson, Charles R (1985). "Matrix Analysis". Section 7.3. Cambridge University Press. ISBN 0521386322.
 Horn, Roger A. and Johnson, Charles R (1991). Topics in Matrix Analysis, Chapter 3. Cambridge University Press. ISBN 0521467136.
 Strang G (1998). "Introduction to Linear Algebra". Section 6.7. 3rd ed., WellesleyCambridge Press. ISBN 0961408855.
One classification: matrix decomposition
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