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	Skew-symmetric matrix - Wikipedia, the free encyclopedia
		In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:
		The "skew-symmetric component" of a matrix A is the matrix B = (A − A
		The eigenvalues of a skew-symmetric matrix always come in pairs ±λ (except in the odd-dimensional case where there is an additional unpaired 0 eigenvalue).
	en.wikipedia.org /wiki/Skew-symmetric_matrix (579 words)

	Symmetric matrix - Wikipedia, the free encyclopedia
		In linear algebra, a symmetric matrix is a matrix that is its own transpose.
		Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.
		Every square real matrix X can be written in a unique way as the sum of a symmetric and a skew-symmetric matrix (a matrix is skew-symmetric if it equals the negative of its transpose).
	en.wikipedia.org /wiki/Symmetric_matrix (515 words)

	Matrix Reference Manual: Special Matrices
		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.
		A Rotation matrix is orthogonal with a determinant of +1.
		A signature matrix is a diagonal matrix whose diagonal entries are all +1 or -1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4190 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-13)
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
		the inverse of the downshift matrix is the upshift matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13731 words)

	Comments on M.W. Evans' "Duality and the Antisymmetric Metric"
		(i) The existence of an "antisymmetric metric", which MWE based on the claim that an antisymmetric matrix should be able to replace the usual symmetric matrix of differential geometry.
		Gravitation therefore is a manifestation of curved spacetime with a symmetric metric, and electromagnetism is a manifestation of spinning spacetime with an antisymmetric metric.
		In general curvilinear coordinates both the symmetric and antisymmetric metrics are defined in terms of the same set of scale factors, and this result can be used in principle to measure the effect of one field on the other.
	www.mathematik.tu-darmstadt.de /~bruhn/Comment-Chap2.htm (2570 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		The point of this exercise is that (verify this!) that powers of the matrix Q have eigenvalues which are the corresponding powers of the eigenvalues of Q. Thus, the power sum tj has a direct interpretation in terms of the trace of Q^j, and this trace in turn has a geometric interpretation.
		tr(Q* R) Note that for an n by n invertible matrix G, the vector space of n by n matrices, M(n), is the orthogonal direct sum of the skew-adjoint matrices and the self-adjoint matrices.
		Exercise: verify that on R^(2n), the Sp(n) adjoint of a matrix is [ A B ]* [ D' -B' ] [ C D ] = [ -C' A' ] where again the matrix has divided into blocks in the obvious way.
	math.ucr.edu /home/baez/PUB/invariance (2967 words)

	Euclidean Tensors
		The matrix that represents it is the unit matrix, of course.
		The 23 component of the antisymmetric tensor is the same as the 1 component of c, for example.
		If the components of an antisymmetric tensor are the infinitesimal velocities of points of a fluid relative to a certain point, then the associated vector is twice the infinitesimal angle of angular velocity of rotation, the vorticity, and is a pseudovector.
	www.du.edu /~jcalvert/math/eucltens.htm (5403 words)

	[No title]
		In a further part of Chapter 2 of his forthcoming book on the GUFT Evans gives an antisymmetric matrix (2.49) that should define a metric eqivalent that one defined by the symmetric matrix (2.42).
		The matrix (2.42) cannot be the metric tensor of spacetime.
		) = diag(−1, 1, 1, 1) denotes the Minkowski matrix.
	www.mathematik.tu-darmstadt.de /~bruhn/MWEsErrors.html (2164 words)

	Creation of Matrices
		A matrix or a vector may also be created by coercing a sequence of ring elements into the appropriate parent matrix structure.
		The sequence Q may be a sequence of m sequences, each of length n and having entries in a ring S, in which case the rows of the matrix are specified by the inner sequences.
		Given a matrix A of any type, return the same matrix but having as parent the appropriate matrix algebra if A is square, or the appropriate R-matrix space otherwise.
	www.umich.edu /~gpcc/scs/magma/text742.htm (1743 words)

	Paper
		Hence, when an arbitrary matrix is multiplied by J from the right-hand (left-hand) side, the order of its columns (rows) is flipped.
		However, the matrix on the left-hand side of this equation is antisymmetric, while that on the right-hand side is symmetric; therefore, matrix
		In the present considerations, it is enough to consider only pairs of degenerate blocks; had the higher degeneracies of the Jordan blocks occurred, one could have considered one pair after another, and at each step one could reduce the dimension of the problem.
	info.fuw.edu.pl /~dobaczew/robled17w/node1.html (2001 words)

	SOBRE LA FUNCIÓMON EXPONECIAL DE UNA MATRIZ, (Site not responding. Last check: 2007-10-13)
		This is motivated by the integration of Frenet–Serret formulae for the case of constant curvatures, and by the relativistic motion of a point charge into a homogeneous electromagnetic field, because in such situations it is necessary to calculate
		coincides with a matrix of Pauli, with application thus to angular momentum in quantum mechanics and in classical mechanics to rigid body dynamics [8].
		Synge [1] proved that the worldline of a charge –under the action of a homogeneous electromagnetic field– has constant curvatures, that is, it represents a helix in Minkowski spacetime.
	wbabin.net /bonilla/matrix.htm (953 words)

	Linear Algebra Glossary
		A nonsingular integer matrix is in Hermite Normal Form if it is lower triangular, all entries are non-negative, and each row has a unique maximum element which is located on the main diagonal.
		Eigenvalue programs typically transform a matrix into upper Hessenberg form, and then carry out the QR method on this matrix, which converges rapidly to a matrix which is diagonal except for 2 by 2 blocks corresponding to complex eigenvalues.
		The identity matrix, usually denoted I, is a square matrix with 1's on the main diagonal and 0's elsewhere.
	orion.math.iastate.edu /burkardt/papers/linear_glossary.html (13535 words)

	Matrix (Site not responding. Last check: 2007-10-13)
		Conversely, every matrix corresponds to a unique linear transformation.
		The matrix is an important concept in mathematics, and was first formulated by
		Matrix Multiplication) together to yield a new matrix.
	www.math.sdu.edu.cn /mathency/math/m/m092.htm (84 words)

	[No title]
		class mat_anti; // antisymmetric matrix // auxiliary classes (compute static values at compile time) template
		// // for SZxSZ symmetric matrix: // (vector index) = tmap_fwd
		{ // template recursion stop public: // base case: 1x1 symmetric matrix (A and B should both be 0) static const uint N=0; static bool verify(ostream& f) { f
	www.ugcs.caltech.edu /~bruce/graphics/mcrt/geometry/tmap.hh (630 words)

	PDL::LinICA -- Linear Independent Component Analysis (ICA) for PerlDL (Site not responding. Last check: 2007-10-13)
		The package first does a full PCA analysis by finding the eigenvalues and -vectors of the covariance matrix and transforming the data so that its covariance matrix becomes diagonal.
		After this a rotation matrix parametrized by an antisymmetric matrix is optimized to yield the lowest values of diagonal cumulants.
		The package includes some functions to test the resulting alleged statistical independence between any components of your choosing.
	pdl.perl.org /HtmlDocs/ICA.html (231 words)

	Symmetric split (Site not responding. Last check: 2007-10-13)
		Symmetric split is an easy exercise on matrices: one knows that any square matrix C of size n×n is the sum of a symmetric matrix A and an antisymmetric matrix B.
		Recall that a square matrix M is symmetric if it equals its transpose.
		Description: write a given matrix as sum of symmetric and antisymmetric matrices.
	wims.unice.fr /wims/en_U1~algebra~symsplit.en.html (181 words)

	International Journal of Applied Mechanics and Engineering
		This paper presents a residual stress analysis and plastic zone behavior in antisymmetric aluminum metal-matrix laminated plates.
		Stainless steel fiber reinforced aluminum metal-matrix laminated plates of constant thickness are formed by stacking four layers bonded antisymmetrically.
		It is assumed that the laminated plates are subjected to transverse uniform loads.
	www.ijame.uz.zgora.pl /ijame_files/archives/v6n1.htm (2360 words)

	Aldor Compiler User Guide: Section III, Chapter 18: Using Aldor with AXIOM
		antisymmetricPart : Matrix R -> Matrix R ++ `antisymmetricPart(M)' returns an antisymmetric ++ matrix `A', computed as `(M - transpose M)/2'.
		It is possible to use compiled Aldor code in subsequent AXIOM sessions.
		A number of input files giving examples of using Aldor with AXIOM are distributed with version 2.0 of the AXIOM system.
	www.aldor.org /docs/HTML/chap18.html (5299 words)

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