Multilinear Maps: Exploring Vectors, Fields & Beyond

[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.

 

[blue_raver22]

I can provide some insights into the topic of multilinear maps and the potential applications in various fields.

Firstly, multilinear maps are a generalization of bilinear maps, which are extensively studied in the theory of quadratic forms and hermitian forms. These forms have important applications in areas such as physics, engineering, and computer science. For example, they are used in quantum mechanics to describe the interactions between particles, in signal processing to analyze data, and in computer graphics to model geometric transformations.

Multilinear maps, on the other hand, have a wider range of applications due to their ability to handle more than two vector spaces. They can be used to represent complex relationships between multiple variables, making them useful in areas such as machine learning, data analysis, and optimization. For instance, multilinear maps are used in machine learning algorithms to capture the interactions between multiple features in a dataset, leading to more accurate predictions.

Furthermore, multilinear maps have connections to other areas of mathematics such as tensor analysis and exterior algebra. These connections allow for the application of multilinear maps in differential geometry, topology, and algebraic geometry. They also have important applications in theoretical physics, particularly in the study of spacetime and general relativity.

In terms of books and resources, there are several texts that cover multilinear maps and their applications. Some examples include "Multilinear Algebra" by Werner Greub, "Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers" by Hung Nguyen-Schäfer, and "Multilinear Functions of Direction and Their Uses in Differential Geometry" by David Hestenes.

In summary, multilinear maps are a powerful tool in mathematics with a wide range of applications in various fields. Their ability to handle multiple variables and capture complex relationships make them valuable in areas such as machine learning, data analysis, and theoretical physics.