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...

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

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

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...

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...

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

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

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...

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 ⊗...

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

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...

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...

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

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...

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...

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 ) ↦...

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...

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...

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...

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...

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

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...

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...

associated with a positive definite kernel. In multilinear subspace learning, PCA is generalized to multilinear PCA (MPCA) that extracts features directly...