Editing Cross product (section)

=== Multilinear algebra ===
In the context of [[multilinear algebra]], the cross product can be seen as the (1,2)-tensor (a [[mixed tensor]], specifically a [[bilinear map]]) obtained from the 3-dimensional [[volume form]],<ref group="note">By a volume form one means a function that takes in ''n'' vectors and gives out a scalar, the volume of the [[Parallelepiped#Parallelotope|parallelotope]] defined by the vectors: <math> V\times \cdots \times V \to \mathbf{R}.</math> This is an ''n''-ary multilinear skew-symmetric form. In the presence of a basis, such as on <math>\mathbf{R}^n,</math> this is given by the determinant, but in an abstract vector space, this is added structure. In terms of [[G-structure|''G''-structures]], a volume form is an [[Special linear group|<math> SL</math>]]-structure.</ref> a (0,3)-tensor, by [[Raising and lowering indices|raising an index]].

In detail, the 3-dimensional volume form defines a product <math> V \times V \times V \to \mathbf{R},</math> by taking the determinant of the matrix given by these 3 vectors. By [[Dual space|duality]], this is equivalent to a function <math> V \times V \to V^*,</math> (fixing any two inputs gives a function <math> V \to \mathbf{R}</math> by evaluating on the third input) and in the presence of an [[inner product]] (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism <math> V \to V^*,</math> and thus this yields a map <math> V \times V \to V,</math> which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".

Translating the above algebra into geometry, the function "volume of the parallelepiped defined by <math> (a,b,-)</math>" (where the first two vectors are fixed and the last is an input), which defines a function <math> V \to \mathbf{R}</math>, can be ''represented'' uniquely as the dot product with a vector: this vector is the cross product <math> a \times b.</math> From this perspective, the cross product is ''defined'' by the [[scalar triple product]], <math>\mathrm{Vol}(a,b,c) = (a\times b)\cdot c.</math>

In the same way, in higher dimensions one may define generalized cross products by raising indices of the ''n''-dimensional volume form, which is a <math> (0,n)</math>-tensor.
The most direct generalizations of the cross product are to define either:
* a <math> (1,n-1)</math>-tensor, which takes as input <math> n-1</math> vectors, and gives as output 1 vector – an <math> (n-1)</math>-ary vector-valued product, or
* a <math> (n-2,2)</math>-tensor, which takes as input 2 vectors and gives as output [[skew-symmetric tensor]] of rank {{nowrap|''n'' − 2}} – a binary product with rank {{nowrap|''n'' − 2}} tensor values.  One can also define <math>(k,n-k)</math>-tensors for other ''k''.

These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and [[parity (physics)|parity]].

The <math> (n-1)</math>-ary product can be described as follows: given <math> n-1</math> vectors <math> v_1,\dots,v_{n-1}</math> in <math>\mathbf{R}^n,</math> define their generalized cross product <math> v_n = v_1 \times \cdots \times v_{n-1}</math> as:
* perpendicular to the hyperplane defined by the <math> v_i,</math>
* magnitude is the volume of the parallelotope defined by the <math> v_i,</math> which can be computed as the Gram determinant of the <math> v_i,</math>
* oriented so that <math> v_1,\dots,v_n</math> is positively oriented.
This is the unique multilinear, alternating product which evaluates to <math> e_1 \times \cdots \times e_{n-1} = e_n</math>, <math> e_2 \times \cdots \times e_n = e_1,</math> and so forth for cyclic permutations of indices.

In coordinates, one can give a formula for this <math> (n-1)</math>-ary analogue of the cross product in '''R'''<sup>''n''</sup> by:

:<math>\bigwedge_{i=0}^{n-1}\mathbf{v}_i =
  \begin{vmatrix}
    v_1{}^1 &\cdots &v_1{}^{n}\\
    \vdots  &\ddots &\vdots\\
    v_{n-1}{}^1 & \cdots &v_{n-1}{}^{n}\\
    \mathbf{e}_1 &\cdots &\mathbf{e}_{n}
  \end{vmatrix}.
</math>

This formula is identical in structure to the determinant formula for the normal cross product in '''R'''<sup>3</sup> except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors ('''v'''<sub>1</sub>, ..., '''v'''<sub>''n''−1</sub>, Λ{{su|b=i=0|p=''n''–1}}'''v'''<sub>''i''</sub>) have a positive [[orientation (mathematics)|orientation]] with respect to ('''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>). If ''n'' is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that ''n'' is even, however, the distinction must be kept. This <math> (n-1)</math>-ary form enjoys many of the same properties as the vector cross product: it is [[alternating form|alternating]] and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments. Moreover, the product <math>[v_1,\ldots,v_n]:=\bigwedge_{i=0}^n v_i</math> satisfies the Filippov identity,
:<math>
[[x_1,\ldots,x_n],y_2,\ldots,y_n]] = \sum_{i=1}^n [x_1,\ldots,x_{i-1},[x_i,y_2,\ldots,y_n],x_{i+1},\ldots,x_n],
</math>
and so it endows '''R'''<sup>n+1</sup> with a structure of n-Lie algebra (see Proposition 1 of <ref>{{cite journal |last1=Filippov |first1=V.T. |date=1985 |title=n-Lie algebras |url=https://link.springer.com/article/10.1007/BF00969110 |journal=Sibirsk. Mat. Zh. |volume=26 |issue=6 |pages=879–891 |doi=10.1007/BF00969110 |bibcode=1985SibMJ..26..879F |s2cid=125051596 |access-date=}}</ref>).