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Topic: Matrix decomposition

Related Topics

Block LU decomposition

LU decomposition

Crout decomposition

Lower triangular matrix

List of linear algebra topics

Diagonal matrix

Cholesky decomposition

Singular value decomposition

Unitary matrix

Invertible matrix

Pseudoinverse

Conjugate transpose

Glasgow Subway

Strathclyde Passenger Transport

Alfred Nobel

	NationMaster - Encyclopedia: Matrix decomposition
		The rest of the matrix is an n×n orthogonal matrix; thus O(n) is a subgroup of O(n+1) (and of all higher groups).
		The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.
		A matrix of dimension m by 1 is called a column vector; a matrix of dimension 1 by n is called a row vector.
	www.nationmaster.com /encyclopedia/Matrix-decomposition (348 words)

	Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition
		Thus, an identity matrix is a diagonal matrix with 1's along its diagonal and zeros elsewhere.
		Matrix addition is commutative, since A + B = B + A. Subtraction of Matrices.
		The actual L matrix is obtained from the RHS matrix of the final tableau by placing ones along its diagonal.
	www.egwald.com /linearalgebra/matrices.php (3984 words)

	Orthogonal matrix - Wikipedia, the free encyclopedia
		Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) absolute value 1.
		The rest of the matrix is an n×n orthogonal matrix; thus O(n) is a subgroup of O(n+1) (and of all higher groups).
		The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.
	en.wikipedia.org /wiki/Orthogonal_matrix (2751 words)

	Method and apparatus for circuit simulation using parallel processors including memory arrangements and matrix ...
		The matrix is used to solve a number of simultaneous equations for the variables, for example, voltages, at each node within the circuit.
		A further feature of the present invention relates to synchronizing the parallel processors during matrix decomposition by assigning a single processor to a given row within the matrix and setting a flag when the values in a row extending from a diagonal matrix element are ready for use.
		In the present invention locking the entire matrix, or even some portion thereof, is avoided by the utilization of a memory structure which accommodates the storage of the individual contributions to a given matrix location entry from each of the circuit elements affecting that matrix location entry.
	www.freepatentsonline.com /5157778.html (4302 words)

	Matrix Decomposition Methods
		Performs LU decomposition on a square matrix, i.e., the matrix A is decomposed as,
		On decomposition, the matrix A is replaced by both L and U. The new changed row index is returned in the valarray ri for performing the back substitution later.
		The matrix is inverted in-place using minimum memory, i.e., the original matrix is replaced by its inverse.
	www.techsoftpl.com /matrix/doc/matdcmp.htm (995 words)

	CHAPTER 6. TIME SERIES DECOMPOSITION
		A fifth series, written to column 5 in the decomposition matrix, is generated by computing the ratios of entries in column 3 (SxI) to the corresponding entries in column 4 (S), which by cancellation (or division) yields an isolated irregular, I, series.
		Finally, a seventh series, written to column 7 of the decomposition matrix, is generated by computing the ratios of the entries in the second column (TxC) by the corresponding entries in the sixth column (T), which by cancellation yields an isolated cyclical, C, Series.
		Given the data in columns 4 through 7 of the decomposition matrix, the seasonal adjustment factors computed from column 5, and the trend (or multiple) regression equation which generated column 6, a forecast may be constructed.
	facweb.furman.edu /~dstanford/forecast/h6.htm (1628 words)

	Computation of the Rotation Matrix
		In motion analysis, it is often necessary to compute the transformation matrix directly from the coordinates of the markers fixed on a moving body.
		Once the transformation matrix is known, the orientation angles and eventually the location of center of rotation as well may be computed.
		In addition, matrix c' must be orthogonal because all three matrices composing c' are orthogonal.
	kwon3d.com /theory/jkinem/rotmat.html (415 words)

	[No title]
		Matrix A = U; Matrix A = U * L; Only conversions that don't lose information are supported - eg you cannot convert an upper triangular matrix into a diagonal matrix using =.
		The dimensions of the matrix on the left hand side are adjusted to those of the matrix or expression on the right hand side.
		Eigenvalues ----------- An eigenvalue decomposition of a symmetric matrix A is a decomposition A = V * D * V.t() where V is an orthogonal matrix and D is a diagonal matrix.
	nersp.nerdc.ufl.edu /~cgs2422/cpp/files/newmat/newmat.txt (2835 words)

	Lab Exercises and Examples
		It is a simple array decomposition used to demonstrate the distribution of data among multiple tasks and the communications required to accomplish it.
		A 512x512 complex matrix is initialized with a point source and then decomposed and distributed to each processor in the partition (by rows for C; by columns for Fortran).
		Finally, the columns/rows of the matrix are gathered back at the destination processor and timing and Mflop results are displayed.
	www.mhpcc.edu /training/workshop/mpi/code_descrip.html (812 words)

	Citations: Rank and null space calculations using matrix decomposition without column interchanges - Foster ...
		Both column and row interchanges must be used in order to preserve better the sparsity of matrix A during the orthogonal decomposition.
		Recently, there has been an increased interest in updating techniques for the SVD and URV decomposition, which converge to the exact SVD or URV under certain stationarity conditions [6, 7] It should be noted that all these decompositions require O(am 2 n) operations, for an m n matrix,....
		Hankel and Toeplitz matrices) RRQR and UTV decompositions are also alternatives to the SVD, but the situation is more complicated here where the initial triangular factorization algorithm is of complexity O(mn)....
	citeseer.ist.psu.edu /context/57549/0 (1768 words)

	BigMatrixImpl (Math 1.2-dev API)
		Returns the trace of the matrix (the sum of the elements on the main diagonal).
		Returns a fresh copy of the cached LU matrix if this has been computed; otherwise the composition is computed and cached for use by other methods.
		The matrix returned is a compact representation of the LU decomposition.
	jakarta.apache.org /commons/math/apidocs/org/apache/commons/math/linear/BigMatrixImpl.html (1475 words)

	Linear Algebra - Preliminary lectures (Site not responding. Last check: )
		A block matrix is a matrix which can be divided into similar parts i.e.
		Since the matrix corresponding to the linear transform is diagonal and since each eigenvector corresponds to a dimension (i.e.
		is an orthogonal matrix is a matrix square root of
	vision.unige.ch /~marchand/teaching/linalg/linalg.html (1819 words)

	Base and Strong Generator Functions
		Construct a presentation for the matrix group G on a set of strong generators and return the presentation in the form of a finitely presented group F that is isomorphic to G. In Magma, the Todd-Coxeter Schreier algorithm is used to construct the presentation.
		The lengths of the basic orbits as defined by the current base for the matrix group G. This function assumes that a BSGS is known for G. The lengths are returned as a sequence of integers.
		Given a matrix group G for which a base and strong generating set are known, and an integer i, where 1 <= i <= k with k the length of the base, return the subgroup of G which fixes the first i - 1 points of the base.
	www.math.uiuc.edu /Software/magma/text269.html (3325 words)

	video summarization via similarity matrix factorization
		The singular values are used to determine the effective rank of the matrix, which corresponds to the number of significant segment clusters in the source video.
		We use the singular vectors to decompose the similarity matrix into a matrix sum of rank-one terms, each of which correspond to one of the segment clusters.
		This linear matrix decomposition is designed to build a purely additive, or parts-based matrix sum, approximating the original similarity matrix.
	www.fxpal.com /?p=vidSumNMF&print=1 (415 words)

	A Grasp for a Matrix Decomposition Problem in Traffic Assignment (Site not responding. Last check: )
		Given a traffic matrix, we search for its decomposition into a sum of matrices with at most one element per row and per column, such that the sum of the largest element of each matrix appearing in the decomposition is minimized.
		The procedure is re-applied to the remaining non-assigned elements, until all entries of the original matrix are assigned to exactly one commutation mode matrix.
		The neighborhood definition is based on the insetion/elimination of some elements appearing in one the matrices, followed by the reconstruction of a new decomposition.
	dmawww.epfl.ch /roso.mosaic/ismp97/ismp_abs_557.html (331 words)

	Looping Constructs
		In the method of LU decomposition, matrix "A" is decomposed into a product of lower and upper triangular matrices.
		Matrix functions are provided for the LU decomposition and forward/backward substitution.
		In the method of LU decomposition, solutions to multiple right-hand vectors can be computed once the matrix [A] has been decomposed.
	www.isr.umd.edu /~austin/aladdin.d/matrix-solve.html (682 words)

	Calculating Orientations
		Since the orientations in a connected component of the interference graph are all relative to one another, we can choose one arbitrary decomposition matrix and derive the rest of the decomposition matrices in the component.
		The remaining decompositions in the connected component are calculated in a similar fashion.
		The base case is the array x for which we chose an arbitrary decomposition matrix that has the specified kernel.
	suif.stanford.edu /papers/anderson93/node17.html (271 words)

	LU Decomposition
		Notice that this is a square matrix and is not yet augmented with the vector of constants from the equations.
		In fact nothing is being done to the upper triangular matrix, but we need the 1 in the lower matrix to show that.
		If a matrix A can be decomposed into an LU representation, then A is equal to the product of the lower and upper triangular matrices.
	www.aspire.cs.uah.edu /textbook/LU.html (650 words)

	NMath Matrix User's Guide - Chapter 5. Decompositions
		NMath Matrix includes decomposition classes for constructing and manipulating QR and singular value decompositions of the general matrix types in NMath Core.
		NMath Matrix also provides decomposition server classes that construct instances of the decomposition classes, allowing you greater control over how decomposition is performed.
		By default, this decomposition class performs pivoting-that is, it may move columns in the input matrix to increase the robustness of the calculation.
	www.centerspace.net /doc/NMath/Matrix/user/decompositions.html (128 words)

	PlanetMath: Householder transformation
		, a symmetric and orthogonal matrix of the form
		is the identity matrix and we have used any normalized vector
		Cross-references: upper triangular, transformations, orthogonal, column, matrix, decomposition, column vector, vector, identity matrix, orthogonal matrix, symmetric, QR decomposition
	planetmath.org /encyclopedia/HouseholderMatrix.html (104 words)

	[No title]
		This provides such things as linear system solving, matrix inverse, eigenvalues of symmetric matrices, and singular value decomposition for real and complex matrices.
		The input matrix is filled with eigenvectors on exit (each column is an eigenvector).
		The matrix V contains the elements of V in untransposed form.
	lush.sourceforge.net /lush-manual/bb056b98.html (328 words)

	Optimization Online - Complex Matrix Decomposition and Quadratic Programming
		Abstract: This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space.
		In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang can be strengthened for the complex Hermitian matrices.
		As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of LMI.
	www.optimization-online.org /DB_HTML/2005/07/1179.html (179 words)

	Probability matrix decomposition models (Site not responding. Last check: )
		First, we consider Boolean matrix decomposition, conceptualize it as a latent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models.
		Second, probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models.
		Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is pre.
	www.nici.kun.nl /Publications/1996/9313.html (161 words)

	Singular Value Decomposition (SVD)
		SVD allows one to diagnose the problems in a given matrix and provides numerical answer as well.
		Any m x n matrix a (m >= n) can be written as the product of an m x n column-orthogonal matrix u, an n x n diagonal matrix with positive or zero elements, and the transpose of an n x n orthogonal matrix v:
		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)

	Lossy Compression (Site not responding. Last check: )
		The idea here is to apply a matrix decomposition method known as Singular Value Decomposition(SVD) to an image to extract "enough" information form the image to recreate an image that is "closer" to the original.
		If you have done a course in linear algebra, you may remember several matrix decomposition methods that were discussed in class (if you went to class).
		This is a class created by a project named JAMA (A JAVA Matrix Package) by MathWorks and the National Institute of Standards and Technology (NIST) which has been released to the public domain.
	edbardsley.org /classes/15-211/lab4/lossy.html (2617 words)

	cbdsqr - compute the singular value decomposition bidiagonal matrix B.
		cbdsqr computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices.
		On entry, the elements of E contain the offdiagonal elements of of the bidiagonal matrix whose SVD is desired.
		On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q' * C. C is not referenced if NCC = 0.
	docs.sun.com /source/816-2461/cbdsqr.html (522 words)

	Bayesian inference with probability matrix decomposition models (Site not responding. Last check: )
		Meulders, M., De Boeck, P., Van Mechelen, I., Gelman, A., and Maris, E. Bayesian inference with probability matrix decomposition models.
		Probability Matrix Decomposition models may be used to model observed binary associations between two sets of elements.
		More specifically, to explain observed associations between two elements, it is assumed that B latent Bernoulli variables are realized for each element and that these variables are subsequently mapped into an observed data point according to a prespecified deterministic rule.
	www.nici.kun.nl /Publications/2001/14486.html (145 words)

	BigMatrixImpl xref
		If matrix 36 * data are modified using any of the public setXxx methods, the saved 37 * decomposition is discarded.
		Its only use is 1013 * to force recomputation of the LU decomposition when changes have been 1014 * made to the underlying data using direct array references.
		1192 * 1193 * The matrix returned is a compact representation of the LU decomposition.
	jakarta.apache.org /commons/math/xref/org/apache/commons/math/linear/BigMatrixImpl.html (2241 words)

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