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...

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

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

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...

Moreover, the characterization of the determinant as the unique alternating multilinear map 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...

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...

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 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 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...

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

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 multilinear map [ ⋅ , ⋅ , ⋅ ] : R × R × R → R {\displaystyle [\cdot ,\cdot ,\cdot ]:R\times...

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...

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

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...

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...

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...

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

tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index...

before. Braided vector space Braided Hopf algebra Monoidal category Multilinear algebra Fock space Bourbaki, Nicolas (1989). Algebra I. Chapters 1-3...

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...

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

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...

defined by the Leibniz formula. Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal...