In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same...

algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function...

(commutative) base ring R in which 2 is not a zero divisor is alternating. Alternating multilinear map Exterior algebra Graded-symmetric algebra Supercommutative...

are equal Alternating operator, a multilinear map in algebra Alternating permutation, a type of permutation studied in combinatorics Alternating series,...

Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies det ( I ) = 1 {\displaystyle \det(I)=1} still...

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...

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...

the associated statistics are called U-statistics. Alternating multilinear map – Multilinear map that is 0 whenever arguments are linearly dependent...

{\displaystyle x_{i}} are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary...

universal property of exterior powers, this is equivalently an alternating multilinear map: β p : ⨁ n = 1 k T p M → R . {\displaystyle \beta _{p}\colon...

object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects...

extend the notion of pullback to tensors of arbitrary rank, i.e., to multilinear maps on W taking values in a tensor product of r copies of W, i.e., W ⊗...

differential k {\displaystyle k} -forms. They can be thought of as alternating, multilinear maps on k {\displaystyle k} tangent vectors. For this reason, tangent...

the double dual of M. Bilinear map Category:Bilinear maps Inner product space Linear form Multilinear form Polar space Quadratic form Sesquilinear form System...

{\displaystyle p\in M} , ω ( p ) {\displaystyle \omega (p)} is an alternating multilinear map from ( T p M ) k {\displaystyle (T_{p}M)^{k}} to the real numbers...

{\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...

systems. For a non-associative ring or algebra R, the associator is the multilinear map [ ⋅ , ⋅ , ⋅ ] : R × R × R → R {\displaystyle [\cdot ,\cdot ,\cdot ]:R\times...

z ) {\displaystyle [x,y,z]=(xy)z-x(yz)} . By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The...

minors is given in multilinear algebra, using the wedge product: the k-minors of a matrix are the entries in the k-th exterior power map. If the columns...

definition is a discriminant for a singular point on a scalar valued multilinear map. Cayley's first hyperdeterminant is defined only for hypercubes having...

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...

p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R-multilinear map: E p × E ∗ q → T q − 1 p − 1 , ( X 1 , … , X p , ω 1 , … , ω q ) ↦...

Nijenhuis–Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by A. Nijenhuis and...

theoretical physics, a tensor is antisymmetric or alternating on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset...

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...

properties with determinant and permanent. In particular, the immanant is multilinear in the rows and columns of the matrix; and the immanant is invariant...

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...

Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms...

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...

linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusion map of V...