What Are the Applications of Multilinear Maps in Mathematics and Science?

[sponsoredwalk]

Here is a bilinear map φ of two vector spaces V₁ & V₂ into another vector space N with
respect to the bases B = {e₁,e₂} & B' = {e₁',e₂'}:

φ : V₁ × V₂ → N | (m₁,m₂) ↦ φ(m₁,m₂) = φ(∑ᵢλᵢeᵢ,∑_jμ_je'_j)
_______________________________= φ(λ₁e₁ + λ₂e₂,μ₁e₁' + μ₂e₂') = φ(λ₁e₁ + λ₂e₂,μ₁e₁') + φ(λ₁e₁ + λe₂,μ₂e₂')
_______________________________= φ(λ₁e₁,μ₁e₁') + φ(λ₂e₂,μ₁e₁') + φ(λ₁e₁,μ₂e₂') + φ(λe₂,μ₂e₂')
_______________________________= λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂')
_______________________________= ∑ᵢ∑_jλᵢμ_jφ(eᵢ,e_j')
(Hope that's right!)

Now in a discussion of bilinear (quadratic, hermitian) forms the terms φ(eᵢ,e_j') in ∑ᵢ∑_jλᵢμ_jφ(eᵢ,e_j')
are elements of a field, & there seems to be quite a lot of theory built up for the special
case of N being a field, but thus far I can't really find any discussion of what happens
when the φ(eᵢ,e_j') elements are elements of a general vector space.
Furthermore, the general case of multilinear (n-linear) maps expressed in this way
seem to follow the same pattern of focusing on fields (or rings), e.g. determinants.
The limited discussions of tensors & exterior algebra I've seen also follow this format of
focusing on maps into a field.

Just wondering what there is on maps of the form
φ : V₁ × V₂ × ... × V_n→ N
φ : V₁ × V₂ × ... × V_n→ V₁ × V₂ × ... × V_n
φ : V₁ × V₂ × ... × V_n→ W₁ × W₂ × ... × W_n
φ : V₁ × V₂ × ... × V_n→ W₁ × W₂ × ... × W_m

(all being vector spaces) as in what kind of books there are discussing this, what this would be useful for etc...

 

[Deveno]

i believe that what you are looking for is "the tensor product" which is "the most general bilinear map" (with two vector spaces) or "multilinear map" (with more than two).

 

[sponsoredwalk]

Thanks for the response, so I take it that φ : V₁ × V₂ → N is the "most general bilinear map"
that can be formed because of the "universal property" [that φ is the most general bilinear map because
for any other bilinear map h : V₁ × V₂ → M you have a unique linear map ω : N → M such that h = ω o φ]
which if I understand it correctly says that:
ω o φ : V₁ × V₂ → M | (m₁,m₂) ↦ (ω o φ)(m₁,m₂) = ω[φ(∑ᵢλᵢeᵢ,∑_jμ_je'_j)]
__________________________ = ω[λ₁μ₁φ(e₁,e₁') + λ₂μ₁φ(e₂,e₁') + λ₁μ₂φ(e₁,e₂') + λ₂μ₂φ(e₂,e₂')]
__________________________ = ω[λ₁μ₁φ(e₁,e₁')] + ω[λ₂μ₁φ(e₂,e₁')] + ω[λ₁μ₂φ(e₁,e₂')] + ω[λ₂μ₂φ(e₂,e₂')]
__________________________ = λ₁μ₁ω[φ(e₁,e₁')] + λ₂μ₁ω[φ(e₂,e₁')] + λ₁μ₂ω[φ(e₁,e₂')] + λ₂μ₂ω[φ(e₂,e₂')]
__________________________ = ∑ᵢ∑_jλᵢμ_jω[φ(eᵢ,e_j')]
is linear, which itself is just a fancy way of ensuring that image elements of φ are linear
because they can be mapped to image elements of a separate bilinear map h.

So I take it that when you generalize this to an indexed product of vector spaces the
tensor product that results is also the the most general multilinear map φ : ΠᵢVᵢ→ N?

 

[lavinia]

Now in a discussion of bilinear (quadratic, hermitian) forms the terms φ(eᵢ,e_j') in ∑ᵢ∑_jλᵢμ_jφ(eᵢ,e_j')
are elements of a field, & there seems to be quite a lot of theory built up for the special
case of N being a field, but thus far I can't really find any discussion of what happens
when the φ(eᵢ,e_j') elements are elements of a general vector space.
Furthermore, the general case of multilinear (n-linear) maps expressed in this way
seem to follow the same pattern of focusing on fields (or rings), e.g. determinants.
The limited discussions of tensors & exterior algebra I've seen also follow this format of
focusing on maps into a field.

A vector space by definition has scalars in a field.

Linearity makes sense when vector space is replaced by R-module where R is a commutative ring.