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Topic: Singular matrix

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	PlanetMath: matrix condition number
		The condition number is a measure of stability or sensitivity of a matrix (or the linear system it represents) to numerical operations.
		In other words, we may not be able to trust the results of computations on an ill-conditioned matrix.
		This is version 7 of matrix condition number, born on 2002-09-28, modified 2006-10-07.
	planetmath.org /encyclopedia/MatrixConditionNumber.html (162 words)

	[No title]
		One way to see why this is a nearly singular matrix is because as n becomes larger, the last few rows are closer and closer in numeric value, making the last few rows nearly the same.
		This is the effect of having a nearly singular matrix.
		If a matrix is nearly singular, it is usually a good idea to not use it, or to modify it somehow to avoid using bad information.
	www.cs.wisc.edu /~hasti/cs310/indivLab/LinSys2/LinSys2IndivLab.html?class=MathML (641 words)

	Invertible matrix - Wikipedia, the free encyclopedia
		Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
		A is row-equivalent to the n-by-n identity matrix I
		Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
	en.wikipedia.org /wiki/Invertible_matrix (980 words)

	Method and system for real-time information analysis of textual material - Patent 5559940
		The singular values are arranged along the principal diagonal in the matrix V and describe the magnitude of the associated textual features.
		The left singular vectors, L, correspond to the relative location of the associated textual features within the matrix X. The singular values, V, are displayed in a diagonal form and may be modified to enhance or suppress the importance of selected singular vectors.
		The resulting left singular vectors, L.sub.i, would indicate the relative location of the subject material in the text matrix associated with the topic of interest and would be used to identify a subset of the right singular vectors, S.sub.i.sup.t, which indicates the subject matter of interest.
	www.freepatentsonline.com /5559940.html (5417 words)

	NMath Matrix User's Guide - 5.2 Singular Value Decompositions
		Instances of the singular value decomposition classes are constructed from general matrices of the appropriate datatype.
		By default, the reduced singular value decomposition and all singular vectors are computed.
		Once a singular value decomposition object has been constructed from a matrix, various read-only properties are provided for retrieving the elements of the decomposition, and for retrieving information about the original matrix:
	www.centerspace.net /doc/NMath/Matrix/user/decompositions3.html (690 words)

	Linear Algebra
		harmless for a non-singular matrix A. Hence, it may be performed always, without the necessity of ascertaining the singularity of the matrix A. Now, you have obtained the factorization of A as A = < D >.
		For this matrix A, we have Vl < = Vr >tr = V, where Vl is the left null-space of A, Vr is the right null-space of A, and V is the null-space of D -- the diagonal in the factorization of the matrix A, as A = < D >.
		For an n by n matrix, the degree of this characteristic polynomial is n.
	www.rism.com /LinAlg/real.htm (4596 words)

	GPS ToolKit Software Library: SVD Class Template Reference (Site not responding. Last check: 2007-10-11)
		Given a matrix A [m,n], the SVD of A = UStranspose(V), where U is [m,m], V is [n,n], and S is [m,n] (like A).
		The singular values may be sorted, as the SVD is invariant under a consistent re-ordering of {singular values / columns of U / columns of V}.
		If some of the singular values are zero, the 'generalized inverse' of A may be formed by editing the singular values in this way: if the ratio of S[i] to S[0] (where S[0] is the largest singular value) is bigger than some tolerance (1.e-7 is good), then 1/S[i] is set to zero in the inverse.
	gpstk.sourceforge.net /doxygen/classgpstk_1_1SVD.html (400 words)

	6.3 - The Inverse of a Square Matrix
		The identity matrix that results will be the same size as the matrix A. Wow, there's a lot of similarities there between real numbers and matrices.
		Row-reduce (I suggest using pivoting) the matrix until the left side is the Identity matrix.
		Divide all the values by the determinant of the matrix (since we haven't talked about the determinant, for a 2×2 system, it is the product of the elements on the main diagonal minus the product of the other two elements).
	www.richland.edu /james/lecture/m116/matrices/inverses.html (1467 words)

	Matrix Reference Manual: Special Matrices
		A 2#2 orthogonal matrix is either a Givens rotation or a Givens reflection according to whether it is proper or improper.
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		Every 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether its determinant is +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4464 words)

	About the Singular Value Decomposition
		In texts, the matrix S is a diagonal matrix with the singular values on the main diagonal, and zeroes elsewhere.
		Another property of the singular values is that their product is the determinant of the matrix, if it's a square matrix; this is why at least one of the singular values is zero if the matrix is square and singular.
		If any of the singular values are zero, such as the last one, the corresponding row of the V matrix will give the combination of the columns of A that result in a zero vector for a result.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=90258 (1718 words)

	Singular Value Decomposition (SVD) tutorial
		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.
		If the matrix A is a real matrix, then U and V are also real.
	web.mit.edu /be.400/www/SVD/Singular_Value_Decomposition.htm (598 words)

	Singular value decomposition - ALGLIB
		It reduces a rectangular matrix to a diagonal matrix by using a sequence of elementary rotations.
		The basis of the most popular modern singular value decomposition algorithms lies in the matrix reduction to a bidiagonal form by orthogonal transformation (this problem is sufficiently simple and requires a finite number of operations to solve it) and its diagonalization by using an iterative QR algorithm.
		For instance, the square matrix singular value decomposition by "divide-and-conquer" when N=100 is 2-4 times faster than by a simple QR algorithm (including the time required to reduce the matrix to bidiagonal form), and is 6-7 times faster when N=1000.
	www.alglib.net /matrixops/general/svd.php (1165 words)

	Advanced Topic: Matrix Decompositions
		Singular value decomposition is an important element of many numerical matrix algorithms.
		Singular value decomposition writes any matrix as a product of a diagonal matrix with row and column orthonormal matrices.
		The stability of numerical matrix algorithms is sometimes improved by "pivoting" or "balancing", in which the rows and columns of the matrix
	documents.wolfram.com /v4/MainBook/3.7.10.html (451 words)

	gallery
		Explicit formulas are known for the inverse and determinant of a Cauchy matrix.
		A circulant matrix has the property that each row is obtained from the previous one by cyclically permuting the entries one step forward.
		The matrix is similar to the symmetric tridiagonal matrix with the same diagonal entries and with off-diagonal entries 1, via a similarity transformation with
	grove.ufl.edu /matlab_help/techdoc/ref/gallery.html (1292 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)

	'Singular Gradient Matrix' Errors and Nonlinear Re - Knowledge Base from Adept Scientific - The Technical Computing ...
		The Nonlinear Regression dialog relies on the function 'nls()' for its main algorithm, and this is the function that is referred to in the discussion.
		It is explained below that the "singular gradient" error is usually eliminated by choosing different initial values for the model parameters, or if you have already tried this, it may be possible that your nonlinear model is singular in many points surrounding the optimal parameter values.
		where H(x) is the Hessian matrix of the sum of squares of the residuals r(x).
	www.adeptscience.co.uk /kb/article/20D2 (805 words)

	PDL::MatrixOps -- Some Useful Matrix Operations
		Matrix manipulation, particularly with large matrices, is a challenging field and no one algorithm is suitable in all cases.
		If you want to use threading on a matrix that's less than, say, 10x10, and might be near singular, then you might want to use determinant, which is a more robust (but slower) determinant finder, instead.
		LU decomposition is the answer to a lot of matrix questions, including inversion and determinant-finding, and lu_decomp is used by inverse.
	pdl.sourceforge.net /PDLdocs/MatrixOps.html (2506 words)

	2 x 2 Singular Matrices (Site not responding. Last check: 2007-10-11)
		After learning what a singular matrix was, we then went on to study the invertible matrices leaving the singular ones behind.
		A matrix is said to be singular if and only if it has no inverse with respect to the identity matrix and matrix multiplication.
		The definition of singular matrices always seemed to imply they weren’t special and often they were ignored for the rest of the course.
	www.ohioriverpottery.com /sec1.html (226 words)

	Introduction to Econometrics - Matrix Algebra
		A special diagonal matrix is a matrix where all diagonal elements are equal to one (identity matrix denoted I).
		If A is a non singular square matrix then the roots of the inverse of A are equal to the reciprocal values of the roots of A but the eigenvectors are the same.
		The determinant of an orthogonal matrix is equal to 1 or -1.
	www.xycoon.com /matrix_algebra.htm (2259 words)

	Positive Definite, Positive Semidefinite Covariance Matrix
		Because a correlation matrix is essentially a normalized covariance matrix, results apply equally to them.
		In this sense, a singular covariance matrix indicates that at least one component of a random vector is extraneous.
		Cholesky matrix A lower-triangular matrix that acts as a matrix "square root" for a positive definite matrix.
	www.riskglossary.com /link/positive_definite_matrix.htm (996 words)

	Algorithm of Singular Value Decomposition
		VI Add to the matrix U the rest of m-r vectors using the Gram-Schmidt orthogonalization process.
		Thus the singular value decomposition of the matrix A is
		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)

	Singular value decomposition and principal component analysis
		In both cases, the n columns of the gene expression data matrix X correspond to assays, and the m rows correspond to the genes.
		One approach is to ignore components beyond where the cumulative relative variance or singular value becomes larger than a certain threshold, usually defined upon the dimensionality of the data.
		As an example of practical use of this kind of SVD-based analysis, it may be possible to detect whether the expression profile of a tissue culture changes in response to radiation dose, even when it is not possible to detect which specific genes change their expression in response to radiation dose.
	public.lanl.gov /mewall/kluwer2002.html (7189 words)

	Matrix Manual: Matrix Decompositions
		The matrix C is the row-echelon or Hermite-normal form of A.
		Every non-singular square matrix A can be expressed as A=PLDU where P is a permutation matrix, L is unit lower triangular, D is diagonal and U is unit upper triangular.
		Every square matrix A can be expressed as A=PLU where P is a permutation matrix, L is a unit lower triangular matrix and U is upper triangular.
	www.psi.toronto.edu /matrix/decomp.html (694 words)

	Assumed knowledge of matrices and vector spaces
		Use of terms singular, diagonal, unit, null, symmetric.
		The transpose of an orthogonal matrix is also orthogonal.
		is a diagonal matrix whose elements are the eigenvalues of
	mcs.une.edu.au /~stat354/notes/node13.html (244 words)

	Error 1313. Decomposition is not possible for a singular matrix. (Site not responding. Last check: 2007-10-11)
		Decomposition is not possible for a singular matrix.
		You have tried to use the LU method with a singular matrix.
		Use error trapping in order to detect a singular matrix while executing LU method.
	www.bluebit.gr /matrix/version_31/Error1313.htm (42 words)

	If a matrix has an inverse, it is said to be invertible or non-singular (Site not responding. Last check: 2007-10-11)
		If a matrix has an inverse, it is said to be invertible or non-singular.
		Reminder that if a matrix has an inverse, then the product of that matrix and its inverse results in the identity matrix (e).
		Since the determinant value for A is equal to -1 and not equal to zero, the inverse of A does exist.
	students.uww.edu /muellerbt15/Inverse.htm (825 words)

	Singular Manual: matrix type cast
		Converts expression to a matrix, where expression must be of type int, intmat, intvec, number, poly, ideal, vector, module, or matrix.
		If int_n and int_m are supplied, then they specify the dimension of the matrix.
		dimensions) of the matrix are determined by the size (resp.
	www.singular.uni-kl.de /Manual/3-0-1/sing_115.htm (90 words)

	Singular Value Decomposition - GNU Scientific Library -- Reference Manual
		In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision.
		For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values.
		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 (448 words)

	Matrix Market: Glossary
		An ill-conditioned matrix is one where the solution to Ax=b is overly sensitive to perturbations in A or b.
		A matrix is totally positive (or negative, or non-negative) if the determinant of every submatrix is positive (or negative, or non-negative).
		A symmetric matrix can always be reduced to a symmetric tridiagonal form by a finite sequence of similarity transformations.
	math.nist.gov /MatrixMarket/glossary.html (756 words)

	Math Forum Discussions
		Singular matrix and non-singular matrix occur in 1907
		Locally, a non-invertible matrix is "singular" in that neighboring
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?threadID=1322827&messageID=4196538 (172 words)

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