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Topic: Bilinear form

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Quadratic form

Positive definite

Equivalence class

Multivariate normal distribution

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	Bilinear operator - Wikipedia, the free encyclopedia
		In mathematics, a bilinear operator is a generalized "multiplication" which satisfies the distributive law.
		The case where X is F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).
		In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear operator V × V → F.
	en.wikipedia.org /wiki/Bilinear_operator (644 words)

	Form - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		Form (Lat forma), in general, refers to the external shape, appearance, configuration of an object, in contrast to the matter or content of which it is composed; thus a speech may contain excellent arguments (the matter may be good), whereas the style, grammar, arrangement (the form) may be bad.
		These forms are not obtained by abstraction from sensible data, nor are they strictly speaking innate: they are obtained by the very action of the mind from the co-ordination of its sensation.
		Also there are forms for taxes; filling one in is a duty to have determined how much tax one owes, and/or the form is a request for a refund.
	encyclopedia.learnthis.info /f/fo/form.html (932 words)

	PlanetMath: symmetric bilinear form (Site not responding. Last check: 2007-11-07)
		A symmetric bilinear form is a bilinear form
		Every inner product over a real vector space is a positive definite symmetric bilinear form.
		This is version 2 of symmetric bilinear form, born on 2002-02-22, modified 2002-04-13.
	planetmath.org /encyclopedia/SymmetricBilinearForm.html (110 words)

	Documentation for Concepts 2.0 (Site not responding. Last check: 2007-11-07)
		A bilinear form a(u,v) is evaluated for two functions of the basis of the space.
		A bilinear form is computed in the application operator of a class derived from concepts::BilinearForm.
		This is done similarly to the bilinear form: the linear form is evaluated for all shape functions of each element.
	www.math.ethz.ch /~concepts/doxygen/html/group__bilinear.html (413 words)

	PlanetMath: bilinear form (Site not responding. Last check: 2007-11-07)
		We note that for a real positive-definite bilinear form that every subspace is non-degenerate, so this gives the usual result about inner product spaces.
		An inner product space on a vector space is a bilinear form if its field is real, but not if it is complex.
		This is version 41 of bilinear form, born on 2002-01-24, modified 2005-01-08.
	planetmath.org /encyclopedia/BilinearForm.html (370 words)

	Documentation for Concepts 2.0 (Site not responding. Last check: 2007-11-07)
		Bilinear form for the Laplace double layer potential with piecewise constant shape functions and hanging nodes (=> recursive subdivision of the larger triangle).
		Bilinear form for the Laplace Single Layer potential with piecewise constant shape functions and hanging nodes (=> recursive subdivision of the larger triangle).
		Bilinear form to compute the adjoint of the Laplace double layer potential.
	www.math.ethz.ch /~concepts/doxygen/html/namespacebem.html (815 words)

	Differential Equation Theory
		Bilinear forms and norms are very intimately related: in a Hilbert space they determine one another.
		It is through the mapping of a norm or the bilinear forms in function space to their counterparts in the boundary value space that the theory or operators on Hilbert space becomes involved in the solutions of differential equations.
		Bilinear forms are mapped from the solution space to the boundary value space in a rather curious way.
	delta.cs.cinvestav.mx /~mcintosh/comun/quant/node4.html (1451 words)

	Univ at Albany: William F. Hammond: Math 220 (Site not responding. Last check: 2007-11-07)
		A symmetric bilinear form on a vector space V is a real-valued function F of two variables from V that satisfies all but the positivity condition in the definition of inner product.
		For every symmetric bilinear form on a finite-dimensional inner product space there is an orthonormal basis of the inner product space in which the matrix of the symmetric bilinear form is diagonal.
		The symmetric bilinear form is itself an inner product if and only if each of the diagonal entries of its matrix, relative to such a basis, is positive.
	nyjm.albany.edu:8000 /~hammond/course/mat220s99/assgt/la990505.html (446 words)

	GAP Manual: 67.18 ClassicalForms
		(that is, an invariant symplectic or unitary bilinear form or an invariant symmetric bilinear form together with an invariant quadratic form, invariant modulo scalars in each case) or try to prove that no such form exists.
		A bilinear form is returned as matrix F such that g * F * g^{tr} equals F modulo scalars for all elements g of
		The "symplectic" indicates that an invariant symplectic form exists, the "unknown" indicates that an invariant "unitary" form might exist.
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C067S018.htm (499 words)

	ipedia.com: Form Article (Site not responding. Last check: 2007-11-07)
		In construction, a form refers to a mold for casting concrete.
		Sometimes part of the form is designed to be permanent, a layer of styrofoam insulation for example.
		Flying forms refer to a type that can be re-positioned to allow a continous casting process.
	www.ipedia.com /form.html (1064 words)

	Decomposition of 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).
		The matrix of the form is stored in bilinearForm and the scalars for each generator of G are stored in scalars.
		If the absolutely irreducible group G preserves an orthogonal form modulo scalars, and so as one component a symmetric bilinear form modulo scalars, this function returns the scalars corresponding to the generators of the group of the symmetric bilinear form.
	www.math.colostate.edu /WWWextra/manuals/magma/htmlhelp/text331.html (3606 words)

	Lie groups (Site not responding. Last check: 2007-11-07)
		Note that any degenerate alternating bilinear form is just a nondegenerate alternating bilinear form on some subspace.
		Note that any degenerate symmetric bilinear form is just a nondegenerate symmetric bilinear form on some subspace.
		Note that any bilinear form is the sum of a symmetric bilinear form and an alternating bilinear form.
	math.ucr.edu /~toby/papers/Lie (810 words)

	Quadratic Forms and Inner Products (Site not responding. Last check: 2007-11-07)
		A bilinear form may be defined on a vector space when it is constructed using the VectorSpace-constructor.
		Given a matrix F belonging to Mat_K(n) that defines a bilinear form on V with respect to the current basis for V, assign F as the bilinear form associated with V. Properties of the Form
		Return the bilinear form on V as a matrix.
	www.math.uiuc.edu /Software/magma/text396.html (289 words)

	Citebase - On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
		Citebase - On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
		On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
		Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9911180 (2047 words)

	A Coercive Bilinear Form for Maxwell's Equations - Costabel (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		A Coercive Bilinear Form for Maxwell's Equations - Costabel (ResearchIndex)
		A Coercive Bilinear Form for Maxwell's Equations (1991)
		Abstract: When one wants to treat the time-harmonic Maxwell equations with variational methods, one has to face the problem that the natural bilinear form is not coercive on the whole Sobolev space H 1.
	citeseer.ist.psu.edu /80768.html (534 words)

	[No title]
		A closely related algebraic operation occurs in the evaluation of multilinear forms such as those that appear in density functional methods for calculating ground electronic states in quantum electronic structures.
		It is tempting to represent them in terms of a precompu ted tensor (a matrix for a bilinear form).
		If one thinks of an integral simply as a very large summation, the analogy between the integral form and the original bilinear form is apparent.
	www.nap.edu /html/mctcc/chap4.18.html (820 words)

	M6221 Lecture Notes 4d (Site not responding. Last check: 2007-11-07)
		Thus B is linear in the first coordinate, and by symmetry is bilinear.
		is a quadratic form, these points lie on a quadric (the hyperbolic quadric).
		Lemma 4.7.3: Let q be a quadratic form of the vector space V, and let Q be the corresponding quadric of P(V).
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc4d.html (725 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		It will be no harder to discuss the invariants of symmetric rank two tensors (bilinear mappings V x V -> R, where V is a finite dimensional real vector space) under the action of GL(V), the group of all linear operators on V, and its subgroups, so we'll immediately pass to this level of generality.
		L' Q L = L^(-1) Q L IOW, the action by O(n) on bilinear forms induced by its action on vectors, is just the conjugation action on matrices M(n).
		More precisely, the form of the function will change since the names of the points change when we change charts, but the same real number will be assigned to a given point.
	math.ucr.edu /home/baez/PUB/invariance (2967 words)

	bilinear
		The bilinear transformation is a mathematical mapping of variables.
		The bilinear transformation maps the s-plane into the z-plane by
		converts to state-space form, performs the transformation, and converts the resulting state-space system back to transfer function form.
	www.weizmann.ac.il /home/info/Matlab5/help/toolbox/signal/bilinear.html (362 words)

	ACTA MATHEMATICA UNIVERSITATIS COMENIANAE (Site not responding. Last check: 2007-11-07)
		Abstract. A bilinear form $f$ on a nonassociative algebra $A$ is said to be invariant iff $f(ab,c) = f(a,bc)$ for all $a,b,c \in A$.
		Finite-dimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure.
		As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel'd but not always isomorphic to a Manin triple.
	www.univie.ac.at /EMIS/journals/AMUC/_vol-66/_no_2/_bordema/bordeman.html (205 words)

	PlanetMath: (Site not responding. Last check: 2007-11-07)
		anti-symmetric bilinear form (=skew-symmetric bilinear form) owned by sleske
		antisymmetric bilinear form (=skew-symmetric bilinear form) owned by sleske
		associative bilinear form (=invariant scalar product) owned by benjaminfjones
	planetmath.org /encyclopedia/A (1936 words)

	Nat' Academies Press, Mathematical Challenges from Theoretical/Computational Chemistry (1995)
		Please use the page image as the authoritative form to ensure accuracy.
		The evaluation of ρ at a single point provides an example of the alternatives for evaluating a bilinear form.
		It is tempting to represent them in terms of a precomputed tensor (a matrix for a bilinear form).
	www.nap.edu /books/0309050979/html/106.html (455 words)

	Documentation for Concepts 2.0 (Site not responding. Last check: 2007-11-07)
		Bilinear form for the stiffness matrix compression with aglomerated wavelets
		Integration classes and basic bilinear form for the eigen value computation in 3d
		Linear form in 2D for the term of Neumann boundary
	www.math.ethz.ch /~concepts/doxygen/html/annotated.html (3432 words)

	cmulB.html (Site not responding. Last check: 2007-11-07)
		Thus, 'cmul' by default computes the Clifford product with the form B while 'cmulB' computes the same product with the user-supplied form B1.
		When used as the third argument, it does not overwrite B. No infix form for 'cmulB' is available since it is intended to be used with the third argument.
		Remember that in that case all three forms B, B1, and B2 must be diagonal.
	math.tntech.edu /rafal/cliff5/Samples/cmulB1.html (241 words)

	Construction of a bilinear antisymmetric form in symplectic vector space (Site not responding. Last check: 2007-11-07)
		Construction of a bilinear antisymmetric form in symplectic vector space
		It contains an axiomatic description of the class of all spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector space over a field F, $\xi: V \times V \to F$ is a bilinear antisymmetric form i.e.
		Construction of a bilinear symmetric form in orthogonal vector space.
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol1/symsp_1.html (241 words)

	No Title (Site not responding. Last check: 2007-11-07)
		For each of the following quadratic forms q, find the corresponding symmetric bilinear form.
		the subspace of symmetric bilinear forms, and B
		Suppose f is a bilinear form on a vector space V, and A is the matrix of f with some basis.
	math.stanford.edu /~rubin/oldcourses/114.w.98/sets/set5/set5.html (138 words)

	The apolar bilinear form in geometric modeling - Abstract (Site not responding. Last check: 2007-11-07)
		Pocchiola and G. Vegter: The apolar bilinear form in CAGD: new applications.
		Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory.
		A generalized version of this inner product was introduced in [1] to derive in a straightforward way some of the recent results in CAGD.
	www.cs.rug.nl /~gert/research/abstracts/st-malo-99.html (124 words)

	The apolar bilinear form in CAGD: new applications (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a xed degree, already known in 19th century invariant theory.
		A generalized version of this inner product was introduced in [9] to derive in a straightforward way some of the recent results in CAGD.
		8 Apolarity and canonical forms for homogeneous polynomials (context) - Ehrenborg, Rota - 1993
	citeseer.ist.psu.edu /237401.html (380 words)

	Documentation for Concepts 2.0 (Site not responding. Last check: 2007-11-07)
		The Jacobian, its inverse and additional quantities only depending on the physical coordinates are only computed once for all components of the bilinear form.
		Index of the bilinear form in the vector valued bilinear form
		Sets up the bilinear form for the div-div part for Maxwell's equation using weighted regularization.
	www.concepts.math.ethz.ch /doxygen/html/classhp2D_1_1DivDiv.html (435 words)

	The apolar bilinear form in geometric modeling - Abstract (Site not responding. Last check: 2007-11-07)
		Vegter: The apolar bilinear form in geometric modeling.
		Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the De Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.
		Keywords: Apolar bilinear form, polarization, homogeneous polynomials, lineal polynomials, dual basis, Euler's identity, Marsden's identity, Bernstein-Bézier patches, B-patches, De Casteljau, De Boor, recurrence relations, algorithm, basis conversion.
	www.cs.rug.nl /~gert/research/abstracts/homogeneous.html (141 words)

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