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- Hyperdeterminant abstract "The hyperdeterminant is a generalization of the determinant in algebra. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes.There are at least three definitions of hyperdeterminant. The first was discovered by Cayley in 1843 (published in 1849). It is usually denoted by det0. The second Cayley hyperdeterminant originated in 1845 and is often called "Det." This definition is a discriminant for a singular point on a scalar valued multilinear map.Cayley's first hyperdeterminant is defined only for hypercubes having an even number of dimensions (although variations exist in odd dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions). The third hyperdeterminant, most recently defined by Glynn (1998), occurs only for fields of prime characteristic p. It is denoted by detp and acts on all hypercubes over such a field.Only the first and third hyperdeterminants are "multiplicative", except for the second hyperdeterminant in the case of "boundary" formats. The first and third hyperdeterminants also have closed formulae as polynomials and therefore their degrees are known, whereas the second one does not appear to have a closed formula or degree in all cases that is known.The notation for determinants can be extended to hyperdeterminants without change or ambiguity. Hence the hyperdeterminant of a hypermatrix A may be written using the vertical bar notation as |A| or as det(A).A standard modern textbook on Cayley's second hyperdeterminant Det (as well as many other results) is "Discriminants, Resultants and Multidimensional Determinants" by Gel'fand, Kapranov, Zelevinsky[2] referred to below as GKZ. Their notation and terminology is followed in the next section.".
- Hyperdeterminant wikiPageExternalLink collectedmathem01caylgoog.
- Hyperdeterminant wikiPageID "15397886".
- Hyperdeterminant wikiPageLength "21287".
- Hyperdeterminant wikiPageOutDegree "39".
- Hyperdeterminant wikiPageRevisionID "674388150".
- Hyperdeterminant wikiPageWikiLink Algebra.
- Hyperdeterminant wikiPageWikiLink Algebraic_geometry.
- Hyperdeterminant wikiPageWikiLink Algebraic_invariant.
- Hyperdeterminant wikiPageWikiLink Andrei_Zelevinsky.
- Hyperdeterminant wikiPageWikiLink Arthur_Cayley.
- Hyperdeterminant wikiPageWikiLink Binet-Cauchy_formula.
- Hyperdeterminant wikiPageWikiLink Category:Multilinear_algebra.
- Hyperdeterminant wikiPageWikiLink Cauchy–Binet_formula.
- Hyperdeterminant wikiPageWikiLink Determinant.
- Hyperdeterminant wikiPageWikiLink Dimension_(vector_space).
- Hyperdeterminant wikiPageWikiLink Discriminant.
- Hyperdeterminant wikiPageWikiLink Dual_space.
- Hyperdeterminant wikiPageWikiLink Einstein_convention.
- Hyperdeterminant wikiPageWikiLink Einstein_notation.
- Hyperdeterminant wikiPageWikiLink Field_(mathematics).
- Hyperdeterminant wikiPageWikiLink Finite-dimensional.
- Hyperdeterminant wikiPageWikiLink Function_(mathematics).
- Hyperdeterminant wikiPageWikiLink Hilberts_basis_theorem.
- Hyperdeterminant wikiPageWikiLink Homogeneous_polynomial.
- Hyperdeterminant wikiPageWikiLink Hypercube.
- Hyperdeterminant wikiPageWikiLink Hypercubes.
- Hyperdeterminant wikiPageWikiLink Hypergeometric_function.
- Hyperdeterminant wikiPageWikiLink Hypergeometric_functions.
- Hyperdeterminant wikiPageWikiLink Invariant_theory.
- Hyperdeterminant wikiPageWikiLink Israel_Gelfand.
- Hyperdeterminant wikiPageWikiLink Least_common_multiple.
- Hyperdeterminant wikiPageWikiLink Levi-Civita_symbol.
- Hyperdeterminant wikiPageWikiLink Multilinear_form.
- Hyperdeterminant wikiPageWikiLink Multilinear_map.
- Hyperdeterminant wikiPageWikiLink Newton_polygon.
- Hyperdeterminant wikiPageWikiLink Newton_polytope.
- Hyperdeterminant wikiPageWikiLink Number_theory.
- Hyperdeterminant wikiPageWikiLink Partial_derivative.
- Hyperdeterminant wikiPageWikiLink Partial_derivatives.
- Hyperdeterminant wikiPageWikiLink Quantum_computing.
- Hyperdeterminant wikiPageWikiLink Quartic.
- Hyperdeterminant wikiPageWikiLink Qubit.
- Hyperdeterminant wikiPageWikiLink Qubits.
- Hyperdeterminant wikiPageWikiLink Ring_(mathematics).
- Hyperdeterminant wikiPageWikiLink Scalar_(mathematics).
- Hyperdeterminant wikiPageWikiLink Simplex.
- Hyperdeterminant wikiPageWikiLink Simplices.
- Hyperdeterminant wikiPageWikiLink Special_linear_group.
- Hyperdeterminant wikiPageWikiLink Square_matrix.
- Hyperdeterminant wikiPageWikiLink String_theory.
- Hyperdeterminant wikiPageWikiLink Tensor.
- Hyperdeterminant wikiPageWikiLink Vector_space.
- Hyperdeterminant wikiPageWikiLink Vector_spaces.
- Hyperdeterminant wikiPageWikiLink Zelevinsky.
- Hyperdeterminant wikiPageWikiLinkText "hyperdeterminant".
- Hyperdeterminant hasPhotoCollection Hyperdeterminant.
- Hyperdeterminant wikiPageUsesTemplate Template:Arxiv.
- Hyperdeterminant wikiPageUsesTemplate Template:Citation.
- Hyperdeterminant wikiPageUsesTemplate Template:Harvs.
- Hyperdeterminant wikiPageUsesTemplate Template:Multiple_issues.
- Hyperdeterminant subject Category:Multilinear_algebra.
- Hyperdeterminant hypernym Generalization.
- Hyperdeterminant type Article.
- Hyperdeterminant type Article.
- Hyperdeterminant type Page.
- Hyperdeterminant comment "The hyperdeterminant is a generalization of the determinant in algebra. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor. Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor.".
- Hyperdeterminant label "Hyperdeterminant".
- Hyperdeterminant sameAs Hiperdeterminante.
- Hyperdeterminant sameAs m.03m71gf.
- Hyperdeterminant sameAs Q5957938.
- Hyperdeterminant sameAs Q5957938.
- Hyperdeterminant wasDerivedFrom Hyperdeterminant?oldid=674388150.
- Hyperdeterminant isPrimaryTopicOf Hyperdeterminant.