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A **multilinear function** (also called a *multilinear form*) is linear in each variable separately (i.e. linear in all its arguments). In other words, it is a linear function of each variable *k* when all the other variables are assigned fixed values. They are polynomials of order* n* but with a maximum order of one for each variable [1].

The end result is that a quantity varies proportionally with several variables, *k*. Bilinear functions (*k* = 2) and trilinear functions (*k* = 3) are special cases.

## Formal Definition of a Multilinear Function

Let’s say we had the following function:

*f*: *W*_{1} * … * *W*_{k} → V

Where *W*_{1}, …, *W*_{k} V are vector spaces over a field 𝔽.

It is a multilinear function if:

For any *w*_{1} ∈ *W*_{1}, *w*_{2} ∈ *W*_{2}… *w*_{n} ∈ *W*_{N} [2].

In other words, it is multilinear if it is homogenously linear in each of its arguments, *k*, separately.

We can also define a multilinear function *f* with matrices and the Kronecker product ⊗ [3]:

*f*(*x*) = α^{T} **m**(**x**),

where α = [α_{1} … α_{2n}]^{T} ∈ ℝ^{2n} is a coefficient row vector and

is a column vector of monomial.

## Multilinear Functions of Vectors

A function of one variable is linear if both of the following statements are true:

*f*(*x*+*y*) = f(*x*) + f(*y*)*f*(*cx*) =*cf*(*c*).

We can build on this statement to define multilinear functions of several vectors. A function of several *vectors* **u**, **v**, … is multilinear if [4]:

*f*(**u**_{1}+**u**_{2},**v**,…) =*f*(**u**_{1},**v**,…) + (**u**_{2},**v**,…),*f*(*c***u**,**v**, …) =*cf*(**u**,**v**,…)*f*(**u**,**v**_{1}+**v**_{2},…) =*f*(**u**_{1},**v**_{1},…) + (**u**,**v**_{2}, …),*f*(**u**,*c***v**, …) =*cf*(**u**,**v**,…)

Multilinear functions of several vectors can easily be built using cross products and dot products. For example: **u**, **v**, **w** → **u** × (**v** × **w**).

## References

[1] Pangalos, G., Eichler, A., & Lichtenberg, G. (2015). Hybrid

multilinear modeling and applications. In Simulation

and Modeling Methodologies, Technologies and Applications (pp. 71-85). Springer, Cham.

[2] 1 Multilinear and alternating functions. 21-801: Algebraic Methods in Combinatorics Multilinear notes. Retrieved May 3, 2021 from: http://www.borisbukh.org/AlgMethods14/multilinear_notes.pdf

[3] Sridharan, A., Lichtenberg, G., Salvador, A. and Salgado, C.

Approaches to Parameter Identification for Hybrid Multilinear Time Invariant Systems. DOI: 10.5220/0009887502550262 In Proceedings of the 10th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2020), pages 255-262. ISBN: 978-989-758-444-2

[4] Dotensko, V. Linear Algebra 1: Lecture 3.

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