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Topic: Singular value decomposition

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Diagonal matrix

Generalized singular value decomposition

Singular matrix

Matrix decomposition

Pseudoinverse

List of linear algebra topics

Unitary matrix

Linear least squares

Conjugate transpose

Symmetric matrix

Tikhonov regularization

Matrix theory

System of linear equations

Eigenvalue

Spectral theorem

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	Singular value decomposition - Wikipedia, the free encyclopedia
		The SVD is also applied extensively to the study of linear inverse problems, and is useful in the analysis of regularization methods such as that of Tikhonov.
		The singular value decomposition was originally developed by differential geometers, who wished to determine whether a real bilinear forms could be made equal to another by independent orthogonal transformations of the two spaces it acts on.
		The first proof of the singular value decomposition for rectangular and complex matrices seems to be by Eckart and Young in 1936; they saw it as a generalization of the principal axis transformation for Hermitian matrices.
	en.wikipedia.org /wiki/Singular_value_decomposition (3174 words)

	SMD : Help : Singular Value Decomposition Help
		SVD is related to the Karhunen-Loeve expansion in the field of pattern recognition and principal component analysis (PCA) in the field of statistics.
		Although the SVD of a dataset is independent of the order of the genes and arrays in the data, a meaningful order might help you correlate dominant eigengenes with experimental artifacts that are superimposed on the data or with biological processes that are present within the data.
		This SVD data normalization, where additive and possibly also multiplicative experimental artifacts are being detected and filtered out, enables better further analysis with methods such as hierarchical clustering, which are sensitive to the presence of an artifact that is superimposed on the data.
	genome-www5.stanford.edu /help/svd.shtml (5089 words)

	Application of singular value decomposition to the analysis of time-resolved macromolecular X-Ray data Biophysical ... (Site not responding. Last check: 2007-10-15)
		ABSTRACT Singular value decomposition (SVD) is a technique commonly used in the analysis of spectroscopic data that both acts as a noise filter and reduces the dimensionality of subsequent least-squares fits.
		To establish the applicability of SVD to crystallographic data, we applied SVD to calculated difference Fourier maps simulating those to be obtained in a time-resolved crystallographic study of photoactive yellow protein.
		SVD takes data-e.g., a set of optical absorption spectra or electron density obtained under different conditions, such as time, pH, or voltage-and represents it by two sets of vectors, which are weighted by their corresponding singular values.
	www.findarticles.com /p/articles/mi_qa3938/is_200303/ai_n9193898 (983 words)

	Singular Value Decomposition
		Singular value decomposition (SVD) is quite possibly the most widely-used multivariate statistical technique used in the atmospheric sciences.
		The purpose of singular value decomposition is to reduce a dataset containing a large number of values to a dataset containing significantly fewer values, but which still contains a large fraction of the variability present in the original data.
		Singular values are equal to the square root of the eigenvalues.
	iridl.ldeo.columbia.edu /dochelp/StatTutorial/SVD (3994 words)

	PlanetMath: singular value decomposition
		SVD can thus be used for rank determination.
		The SVD provides a numerically robust solution to the least-squares problem.
		This is version 5 of singular value decomposition, born on 2002-01-02, modified 2003-01-14.
	planetmath.org /encyclopedia/SingularValueDecomposition.html (90 words)

	Singular value decomposition and principal component analysis
		SVD was used in combination with the spectroscopic data to extract a scattering curve for the collapsed state of the lysozyme, a structural intermediate that was not observed in isolation.
		Singular vectors (in our case eigenassays and eigengenes) whose singular values plot to the right of such an “elbow” are ignored because they are assumed to be mainly due to noise.
		SVD and related methods are particularly valuable analysis methods when the distribution of genes is more complicated than the simple distributions in our example data: for instance, SVD has been used to characterize ring-like distributions of genes such as are observed in scatter plots of cell-cycle gene expression data (Alter et al.
	public.lanl.gov /mewall/kluwer2002.html (7189 words)

	Singular Value Decomposition (SVD) tutorial (Site not responding. Last check: 2007-10-15)
		Singular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is a n x p matrix) in which the n rows represents the genes, and the p columns represents the experimental conditions.
		The SVD represents an expansion of the original data in a coordinate system where the covariance matrix is diagonal.
		The values of x1 and x2 are chosen such that the elements of the S are the square roots of the eigenvalues.
	web.mit.edu /be.400/www/SVD/Singular_Value_Decomposition.htm (598 words)

	Singular Value Decomposition and Its Visualization
		In the context of PCA or SVD, common visualization methods include the biplot [Gabriel, 1971], scatter plots between singular columns or singular rows [Shen and Huang, 2005], etc. The biplot shows the relations between the rows and columns, and the scatter plot can be used to cluster the rows or columns.
		In the FDA field, besides the above plots, it is common to plot singular columns or singular rows as curves [Ramsay and Silverman, 1997].
		Various SVD visualization tools, such as the SVD surface plot, the image plot, the SVD curve movie and the SVD rotation movie, are proposed to find insightful features in a two way data matrix, especially a functional data set.
	www.unc.edu /~lszhang/research/network/SVDmovie/abstractNintro.html (1368 words)

	GNU Scientific Library -- Reference Manual: Singular Value Decomposition
		The singular values \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2 >=...
		The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value.
		The presence of a zero singular value indicates that the matrix is singular.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_206.html (346 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		Among other applications, the SVD can be used to solve under- and overdetermined systems of equations, compress digital images and so on.
		A, then decomposing it into a truncated sum of vector multiplications where only the ten largest singular values are taken into account implies that the amount of data needed to save the image digitally is reduced to ten times 560; a lot less than storing the entire matrix with its 320 times 240 entries.
		An iterative approach to calculating the singular values and the unit vectors is to minimize the error norm using for example the simple
	biphome.spray.se /baxtrom/appliedmath/SVD.htm (510 words)

	Singular Value Decomposition
		In general, the SVD represents an expansion of the original data A in a coordinate system where the covariance matrix is diagonal.
		Consequently, the non-zero singular values of A are
		We use the singular value decomposition algorithm (SVD) to study the magnitude of negative weights and the effect of the correction applied by setting all negative dwell weights equal to 0.
	www.mlahanas.de /Math/svd.htm (2138 words)

	#01005: Computational performance for the SVD (singular-value decomposition) of a very large matrix (Site not responding. Last check: 2007-10-15)
		The scholar is using the MATLAB software package for the SVD (singular-value decomposition) of a very large sparse matrix, that is, an array of 36654 rows and 5937 columns with a sparseness factor of 0.046 percent.
		Instead of non-zero values making up 0.046 percent of the array for the first few singular values, you could be dealing with greater than one percent by the time the algorithm starts on the one hundredth singular value.
		That is, for example, if the first 39 singular values decrease by at most a factor of two or three from one value to the next and the fortieth singular value drops by a factor of 100, the heuristically based decision would be that only the first 39 are significant.
	sunsolve.sun.com /search/document.do?assetkey=1-30-01005-1 (1410 words)

	Singular Value Decomposition (SVD)
		A very powerful set of techniques dealing with sets of equations or matrices that are either singular or numerically very close to singular is the so-called singular value decomposition (SVD).
		SVD allows one to diagnose the problems in a given matrix and provides numerical answer as well.
		The diagonal elements of matrix w are the singular values of matrix a and non-negative numbers.
	kwon3d.com /theory/jkinem/svd.html (243 words)

	The Singular Value Decomposition
		Applications of the Singular Value Decomposition in scientific computing and digital signal processing is an ongoing research theme at IMM, and the activities are lead by Prof.
		shows that each right singular vectors is mapped onto the corresponding left singular vector, and the "magnification factor" is the corresponding singular value.
		The SVD is also an important analytical and computational tool in connection with regularization of inverse problems, with applications in, e.g., computational tomography, image deblurring, and geophysical inversion (seismology).
	www.imm.dtu.dk /~pch/Projekter/tsvd.html (500 words)

	Singular value decomposition (Site not responding. Last check: 2007-10-15)
		Singular value decomposition (SVD) is a general linear-algebraic tool for dealing with singular or near-singular matrices (Noble and Daniel 1977).
		SVD determines the form of S which has minimum length by discarding the singular or near-singular terms in the formal algebraic solution.
		Briggs (1995) briefly discussed the use of SVD for deconvolving VLBA data; he showed that for about 3000 pixel intensities and visibilities, SVD produces an image whose quality is subjectively on a par with that of `CLEAN', but at the cost of large memory use and of long running time (on IBM RS/6000-580 workstations).
	www.cv.nrao.edu /~abridle/deconvol/node28.html (126 words)

	Singular Value Decomposition
		The singular value decomposition (SVD) of A is the factorization
		are called the singular values, the first r columns of V the right singular vectors and the first r columns of U the left singular vectors.
		The SVD may be used to find a minimum norm solution to a (possibly) rank-deficient linear least squares problem (2.1).
	www.netlib.org /lapack/lug/node53.html (581 words)

	The eigen(singular)value decomposition (Site not responding. Last check: 2007-10-15)
		This routine calculates the eigenvalue decomposition of a symmetric semiseparable matrix in the Givens-vector representation.
		SVDUS: The SVD of Upper triangular Semiseparable matrices.
		This routine calculates the singular value decomposition of an upper triangular semiseparable matrix represented with the Givens-vector representation.
	www.cs.kuleuven.ac.be /~raf/homepage/publications/phd/node131.html (57 words)

	Singular Value Decomposition (Site not responding. Last check: 2007-10-15)
		The Singular Value Decomposition (SVD) is a widely used technique to decompose a matrix into several component matrices, exposing many of the useful and interesting properties of the original matrix.
		Using the SVD, we can determine the rank of matrix, quantify the sensitivity of a linear system to numerical error, or obtain an optimal lower-rank approximation to the matrix.
		The mini tutorial on the SVD provides a quick introduction to the definition and properties of the SVD, as well as a listing of some applications in which the SVD is used.
	www.cs.brown.edu /research/ai/dynamics/tutorial/Documents/SingularValueDecomposition.html (210 words)

	Singular Value Decomposition -- A Primer - Leach (ResearchIndex)
		Abstract: Introduction The singular value decomposition (SVD) is a powerful technique in many matrix computations and analyses.
		Using the SVD of a matrix in computations, rather than the original matrix, has the advantage of being more robust to numerical error.
		Additionally, the SVD exposes the geometric structure of a matrix, an important aspect of many matrix calculations.
	citeseer.ist.psu.edu /651803.html (406 words)

	GNU Scientific Library -- Reference Manual - Singular Value Decomposition (Site not responding. Last check: 2007-10-15)
		The singular values @math{\sigma_i = S_{ii}} are all non-negative and are generally chosen to form a non-increasing sequence @math{\sigma_1 >= \sigma_2 >=...
		The singular values are non-negative and form a non-increasing sequence from @math{S_1} to @math{S_N}.
		This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for @math{M>>N}.
	www.math.utah.edu /software/gsl/gsl-ref_206.html (357 words)

	Singular Value Decomposition (SVD)
		Singular value decomposition (SVD) is one of techniques that isolate coupled modes of variability between time series of two fields.
		Their results show that SVD is very simple to perform and interpret and requires no user-supplied parameters.
		Wallace, J.M., C. Smith, and C.S. Bretherton, 1992: Singular Value Decomposition of Wintertime Sea Surface Temperature and 500-md Height Anomalies J. Climate, 5, 561-576.
	daac.gsfc.nasa.gov /precipitation/TDST_SCI/statis_SVD.shtml (152 words)

	The Singular Value Decomposition
		Thus selected singular values and the corresponding right singular vectors may be computed by finding eigenvalues and vectors for the
		The left singular vectors are recovered from the right singular vectors.
		As long as the largest singular values are not multiple or tightly clusterd, there should be no problem in obtaining numerically orthogonal left singular vectors from the computed right singular vectors.
	www.caam.rice.edu /software/ARPACK/UG/node130.html (355 words)

	Algorithm of Singular Value Decomposition
		Thus the singular value decomposition of the matrix A is
		Therefore it is suitable to find nonzero singular values of the matrix A using the
		Applying the singular value decomposition of the matrix A got in example 3.3.3, find the bases of the subspace of the column-vectors
	www.cs.ut.ee /~toomas_l/linalg/lin2/node14.html (421 words)

	Analysis of Metabolic Capabilities Using Singular Value Decomposition of Extreme Pathway Matrices -- Price et al. 84 ...
		The SVD of these three reaction networks demonstrates how the SVD characterizes general properties of the solution cone, defining the phenotypic possibilities of the metabolic networks.
		FIGURE 5 Cumulative fractional contributions for the singular value decompositions of the extreme pathway matrices in H.
		The cumulative fractional contribution is defined as the sum of the first n fractional singular values (reported as a percent).
	www.biophysj.org /cgi/content/full/84/2/794 (5380 words)

	Singular Value Decomposition - GNU Scientific Library -- Reference Manual
		A general rectangular M-by-N matrix A has a singular value decomposition (svd) into the product of an M-by-N orthogonal matrix U, an N-by-N diagonal matrix of singular values S and the transpose of an N-by-N orthogonal square matrix V,
		The diagonal elements of the singular value matrix S are stored in the vector
		The singular values are non-negative and form a non-increasing sequence from S_1 to S_N.
	www.gnu.org /software/gsl/manual/html_node/Singular-Value-Decomposition.html (395 words)

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