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Topic: Symplectic matrix

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	Symplectic manifold
		Fundamental examples of symplectic manifolds are given by the cotangent bundles of manifolds; these arise in classical mechanics, where the set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
		Since the symplectic form on a symplectic manifold is nondegenerate, it sets up an isomorphism between the tangent bundle and the cotangent bundle, thus establishing a one-to-one correspondence between tangent vectors and one-forms.
		A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves; this result has led to the development of a fairly large subdiscipline of symplectic topology.
	pedia.newsfilter.co.uk /wikipedia/s/sy/symplectic_manifold.html (879 words)

	Symplectic matrix - Encyclopedia, History, Geography and Biography
		In mathematics, a symplectic matrix is a 2n×2n matrix M (whose entries are typically either real or complex) satisfying the condition
		Briefly, a symplectic vector space is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric bilinear form ω.
		The condition that f be a symplectic transformation is precisely the condition that M be a symplectic matrix:
	www.arikah.net /encyclopedia/Symplectic_matrix (536 words)

	Symplectic matrix - Wikipedia, the free encyclopedia
		The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space.
		Briefly, a symplectic vector space is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric bilinear form ω.
		The condition that L be a symplectic transformation is precisely the condition that M be a symplectic matrix:
	www.wikipedia.org /wiki/Symplectic_matrix (453 words)

	Encyclopedia: Symplectic matrix
		In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
		Every symplectic matrix has an inverse which is given by In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
		Let M be a 2nÃ—2n block matrix given by In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks.
	www.nationmaster.com /encyclopedia/Symplectic-matrix (1449 words)

	BibTeX bibliography /home/ftp/pub/narep/u-manchester-mccm.bib
		The conditioning of the first companion linearization relative to that of $P$ is shown to depend on the coefficient matrix norms, the eigenvalue, and the left \ev s of the linearization and of $P$.
		We describe a simple procedure to construct a defective matrix whose distance from $A,$ among all the defective matrices, is the shortest.
		Matrix $p$th roots can be computed by a specialized version of Newton's method, but we show that this iteration has poor convergence and stability properties in general.
	www.ma.man.ac.uk /~nareports (6794 words)

	ipedia.com: Symplectic manifold Article (Site not responding. Last check: 2007-09-19)
		A symplectic manifold is a pair where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form.
		A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form.
		Symplectic topology is that part of mathematics concerned with the study of symplectic manifolds.
	www.ipedia.com /symplectic_manifold.html (794 words)

	PlanetMath: symplectic matrix (Site not responding. Last check: 2007-09-19)
		The determinant of a symplectic matrix equals one.
		With standard matrix multiplication, symplectic matrices form a group.
		This is version 2 of symplectic matrix, born on 2003-04-02, modified 2003-04-02.
	planetmath.org /encyclopedia/SymplecticMatrix.html (61 words)

	eLibrary Project : Symplectic group (Site not responding. Last check: 2007-09-19)
		The symplectic group of degree 2''n'' over a field (mathematics),field ''F'', denoted Sp(2''n'', ''F''), is the group (math),group of ''2n'' by ''2n'' symplectic matrix,symplectic matrices with entries in ''F'', and with the group operation that of matrix multiplication.
		Since all symplectic matrices have unit determinant, the symplectic group is a subgroup of the special linear group SL(2''n'', ''F'').
		More abstractly, the symplectic group can be defined as the set of linear transformations of a 2''n''-dimensional vector space over ''F'' that preserve a nondegenerate, skew-symmetric, bilinear form.
	elibraryproject.com /info/symplectic/group.html (365 words)

	Unitary matrix - Wikipedia, the free encyclopedia
		In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition
		A unitary matrix in which all entries are real is the same thing as an orthogonal matrix.
		It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e.
	www.wikipedia.org /wiki/Unitary_matrix (208 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.
		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.
		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 (4204 words)

	An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem
		An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem is presented.
		The Lanczos vectors are constructed to form a symplectic basis.
		The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts.
	epubs.siam.org /sam-bin/dbq/article/34311 (137 words)

	Classical mechanics:Symplectic spaces - Wikibooks
		The reason is the additional property of the phase space (p,q) of having a "symplectic structure".
		This is essentially the matrix generalization of the imaginary unit that we know from complex algebra.
		Simply put, symplectic spaces arise when unit matrices are replaced by symplectic matrices in strategically chosen places.
	en.wikibooks.org /wiki/Classical_mechanics:Symplectic_spaces (200 words)

	Publications
		For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A in G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary.
		On the Determinant of Symplectic Matrices, (with D. Steven Mackey).
		A constructive derivation of the symplectic analogue of the Cartan-Dieudonné theorem is one of the new proofs in this essay.
	homepages.wmich.edu /~mackey/publist.html (1832 words)

	Matrix Groups of Large Degree
		The classical forms are: symplectic (non-degenerate, alternating bilinear), unitary (non-degenerate sesquilinear) or orthogonal (a symmetric bilinear form and a quadratic form).
		If the absolutely irreducible group G preserves a symplectic form (non-degenerate, alternating bilinear) modulo scalars, this function returns the scalars corresponding to the generators of the group of the form.
		This algorithm tries to prove that a matrix or permutation group G is perfect by establishing that its generators are in G'.
	www.math.lsu.edu /magma/text308.htm (5471 words)

	The Parameterized SR Algorithm For Symplectic (Butterfly) Matrices (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		A symplectic matrix B in butterfly form is uniquely determined by 4n \Gamma 1 parameters.
		10.5%: SR and SZ Algorithms for the Symplectic (Butterfly)..
		SR and SZ Algorithms for the Symplectic (Butterfly)..
	citeseer.ist.psu.edu /45003.html (439 words)

	4.2.5 Symplectic Algorithms
		Another quantity that should be conserved in Hamiltonian dynamics is the symplectic form.
		But symplectic integrators conserve a Hamiltonian function that is different from, but close to, the given Hamiltonian (YOSHIDA).
		Apply the (non-symplectic) RK method and the (symplectic) Størmer-Verlet algorithm (or the Candy procedure) to the one-body Kepler problem with elliptic orbit.
	homepage.univie.ac.at /Franz.Vesely/cp0102/dx/node68.html (272 words)

	Symplectic matrix
		In linear algebra, a symplectic maxtrix is a 2n by 2n complex matrix A such that A
		J beeing the 2n by 2n skew-symmetric matrix:
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/sy/Symplectic_matrix.html (56 words)

	Symplectic matrix (Site not responding. Last check: 2007-09-19)
		In mathematics, a symplectic matrix is a 2n by2n matrix M (whose entries aretypically either real or complex) satisfying the condition
		Furthermore, the product of two symplectic matricesis, again, a symplectic matrix.
		It follows easily from the definition that the determinant of anysymplectic matrix is ±1.
	www.therfcc.org /symplectic-matrix-238910.html (218 words)

	Geometry and Mechanics at UniS (Site not responding. Last check: 2007-09-19)
		The research ranges from the theory of symplectic and multi-symplectic geometry to detailed analysis of physical problems in chemistry, physics, optics and fluid mechanics.
		The symplectic Evans matrix, and the instability of solitary waves and fronts with symmetry.
		The central idea is that symplecticity for Hamiltonian PDEs is directional: the symplectic structure of the PDE is decomposed into distinct components representing space and time independently.
	www.maths.surrey.ac.uk /research/Geom (1387 words)

	Symplectic Balancing of Hamiltonian Matrices - Benner (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		It is shown that isolated eigenvalues can be deflated using similarity transformations with symplectic permutation matrices.
		The block reduction of a general matrix to Hessenberg form is based on a block representation of products of Householder transformations...
		11 Balancing a matrix for calculation of eigenvalues and eigenv..
	citeseer.lcs.mit.edu /benner00symplectic.html (591 words)

	SR and SZ Algorithms for the Symplectic (Butterfly) Eigenproblem (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		SR and SZ Algorithms for the Symplectic (Butterfly) Eigenproblem (1992)
		Abstract: SR and SZ algorithms for the symplectic (generalized) eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed.
		A 2n \Theta 2n symplectic butterfly matrix has 8n \Gamma 4 (generically) nonzero entries, which are determined by 4n \Gamma 1 parameters.
	citeseer.ist.psu.edu /15497.html (568 words)

	ENIGMA-RANDOM MATRICES (Site not responding. Last check: 2007-09-19)
		A related class of matrix integrals (in a certain sense the ``holomorphic square roots'' of normal matrix integrals) have recently made their appearance in the string literature, after the seminal paper of R.Dijkgraaf and C.Vafa,and are expected to be important objects in supersymmetric gauge theory and strings.
		The Hermitian matrix integrals are tau-functions for the Toda lattice, whereas Adler-van Moerbeke show the symmetric and symplectic matrix integrals are tau-functions for a new lattice, the Pfaff lattice and also they satisfy a subalgebra of Virasoro constraints.
		Another generalization is to replace the unitary group invariance by orthogonal or symplectic, and further extend the results obtained for the 1-matrix model to all symmetric-spaces ensembles of random matrices.
	enigma.sissa.it /wp2n.html (664 words)

	Analytic Jacobians of Hyperelliptic Curves
		Because we chose a symplectic basis for the homology it follows that tau = omega_2^(-1)omega_1 is an element of Siegel upper half-space.
		The period matrix is computed with respect to the holomorphic differentials varphi_i = x^(i - 1)dx/y and a symplectic basis for the homology that can be retrieved using HomologyBasis.
		Returns the endomorphism ring, as a matrix algebra, of the given analytic Jacobian A. Suppose the analytic Jacobian has big period matrix P. The second return value is a list of alpha-matrices such that alpha P = P M, for each generator, M, of the matrix algebra.
	www.math.lsu.edu /magma/text1245.htm (3528 words)

	Identity matrix -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-19)
		In (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the (The diagonal of a square matrix running from the upper left entry to the lower right entry) main diagonal and zeros elsewhere.
		whenever these (The multiplication of matrices) matrix multiplications are defined.
		The ith column of an identity matrix is the (Click link for more info and facts about unit vector) unit vector e
	www.absoluteastronomy.com /encyclopedia/i/id/identity_matrix.htm (176 words)

	Unitary matrix
		In mathematics, a unitary matrix is a n by n complex matrix
		Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose U
		All eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e.
	www.sciencedaily.com /encyclopedia/unitary_matrix (235 words)

	A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic ... (Site not responding. Last check: 2007-09-19)
		A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices.
		The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic).
		In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.
	archiv.tu-chemnitz.de /pub/1998/0091 (127 words)

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