Let $X$ be a finite set and $\mathbb{F}$ be a field. We say that a function $f: X\times X \to \mathbb{F}$ is rank one if $f$ is of the form $(x,y) \mapsto a(x)b(y)$ for $a,b: X \to \mathbb{F}$. (See here for instance in Tao's set up for the slice rank formulation of the Croot-Lev-Pach polynomial method).
I understand that a matrix $A \in M_{n\times n}(\mathbb{R})$, (for instance) is of rank one if and only if $A = \mathbf{v}\mathbf{w}^t$ for some $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$. However, the connection between the functional definition and the matrix definition has always been unclear to me.
To be explicit about my questions:
- Are we interpreting the function $f: X\times X \to \mathbb{F}$ as a function that, in some sense, "populates" the entries of a $|X| \times |X|$ matrix?
- Can we interpret $f: X^k \to \mathbb{F}$ as a function that "populates" the entries in a $$\underbrace{|X|\times |X| \times \cdots \times |X|}_{k\text{-times}}$$ hypermatrix?
- In the literature, $f$ as defined in (1) and (2) are said to be tensors (see for instance this paper by Gowers). What is the connection between these tensors and the tensors as they are defined in Dummit and Foote (say). I have done some playing around and searching around to try and get a satisfactory answer concerning this connection but have not been very successful. Do you have a hint as to how I can discover this connection? I think I am missing something that is supposedly obvious.
If these questions are too lengthy to answer in a short amount of time, what references would you suggest I look into to gain a better understanding?
Thanks in advance for your time.
A very nice answer to the above question can be found in Chapter 2 of the below monograph:
Tensors: Geometry and Applications by J.M. Landsberg