Interior Product
The interior product is a dual notion of the wedge product in an exterior algebra LambdaV, where V is a vector space. Given an orthonormal basis {e_i} of V, the forms
| {e_(i_1) ^ ... ^ e_(i_p)}_(i_1<...<i_p) |
(1)
|
are an orthonormal basis for Lambda^pV. They define a metric on the exterior algebra, <alpha,beta>. The interior product with a form gamma is the adjoint of the wedge product with gamma. That is,
| <alpha⌟gamma,beta>=<alpha,beta ^ gamma> |
(2)
|
for all beta. For example,
| e_1 ^ e_2⌟e_3=0 |
(3)
|
and
| e_1 ^ e_2 ^ e_3 ^ e_4⌟e_1 ^ e_4=e_2 ^ e_3, |
(4)
|
where the e_i are orthonormal, are two interior products.
An inner product on V gives an isomorphism e:V=V^* with the dual vector space V^*. The interior product is the composition of this isomorphism with tensor contraction.
See also
Exterior Algebra, Inner Product, Tensor Contraction, Wedge ProductThis entry contributed by Todd Rowland
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Rowland, Todd. "Interior Product." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InteriorProduct.html