Multilinear map that is 0 whenever arguments are linearly dependent
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring.
The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space.
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algebra, a multilinearmap is a function of several variables that is linear separately in each variable. More precisely, a multilinearmap is a function...
are equal Alternating operator, a multilinearmap in algebra Alternating permutation, a type of permutation studied in combinatorics Alternating series,...
abstract algebra and multilinear algebra, a multilinear form on a vector space V {\displaystyle V} over a field K {\displaystyle K} is a map f : V k → K {\displaystyle...
(commutative) base ring R in which 2 is not a zero divisor is alternating. Alternatingmultilinearmap Exterior algebra Graded-symmetric algebra Supercommutative...
Moreover, the characterization of the determinant as the unique alternatingmultilinearmap that satisfies det ( I ) = 1 {\displaystyle \det(I)=1} still...
alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal...
object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects...
universal property of exterior powers, this is equivalently an alternatingmultilinearmap: β p : ⨁ n = 1 k T p M → R . {\displaystyle \beta _{p}\colon...
{\displaystyle p\in M} , ω ( p ) {\displaystyle \omega (p)} is an alternatingmultilinearmap from ( T p M ) k + 1 {\displaystyle (T_{p}M)^{k+1}} to the real...
{\displaystyle x_{i}} are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternatingmap is anticommutative. In the binary...
differential k {\displaystyle k} -forms. They can be thought of as alternating, multilinearmaps on k {\displaystyle k} tangent vectors. For this reason, tangent...
{\displaystyle V\odot V} , and the alternating square of V, V ∧ V {\displaystyle V\wedge V} , respectively. The symmetric and alternating squares are also known as...
p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R-multilinearmap: E p × E ∗ q → T q − 1 p − 1 , ( X 1 , … , X p , ω 1 , … , ω q ) ↦...
V; alternating if B(v, v) = 0 for all v in V; skew-symmetric or antisymmetric if B(v, w) = −B(w, v) for all v, w in V; Proposition Every alternating form...
systems. For a non-associative ring or algebra R, the associator is the multilinearmap [ ⋅ , ⋅ , ⋅ ] : R × R × R → R {\displaystyle [\cdot ,\cdot ,\cdot ]:R\times...
v_{1},\dots ,v_{n}} is positively oriented. This is the unique multilinear, alternating product which evaluates to e 1 × ⋯ × e n − 1 = e n {\displaystyle...
z ) {\displaystyle [x,y,z]=(xy)z-x(yz)} . By definition, a multilinearmap is alternating if it vanishes whenever two of its arguments are equal. The...
Pitfall! became the first action game that demanded its fans sit down and map out routes, breaking down the complex arrangement of what initially appears...
definition is a discriminant for a singular point on a scalar valued multilinearmap. Cayley's first hyperdeterminant is defined only for hypercubes having...
associated with a positive definite kernel. In multilinear subspace learning, PCA is generalized to multilinear PCA (MPCA) that extracts features directly...
properties with determinant and permanent. In particular, the immanant is multilinear in the rows and columns of the matrix; and the immanant is invariant...
In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra Λ(V) of a vector space V. This...
before. Braided vector space Braided Hopf algebra Monoidal category Multilinear algebra Fock space Bourbaki, Nicolas (1989). Algebra I. Chapters 1-3...
In multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is...
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