I apologize for the wordiness of this question, but it's largely conceptual (and might not even make sense).
This is a question about Analysis 2 (particularly, analysis on manifolds). I've been having trouble getting an answer to this question.
I know that 2-tensors can be used to represent any matrix, and I know that we have both the set of elementary matrices (those which correspond to the elementary row operations) and the set of elementary tensors. I also know that a linear transformation by matrix multiplication cannot modify the rank of a matrix, whereas linear transformation by a 2-tensor can modify rank.
Would it be correct to say that the set of elementary 2-tensors are a generalization of the set of elementary matrices? By this, I mean to ask if it would be correct to say that the elementary matrices can be represented as elementary tensors, and that 2-tensors can be represented by the union of the set of elementary matrices and the set of elementary matrices with zero rows/columns? That's to say that 2-tensors can do everything the elementary matrices do, but can also modify the rank of a matrix when the linear transformation in consideration is matrix multiplication.
This is an appealing idea, because I'm having trouble understanding the concept of multilinear algebra. In general, is it correct to say that, in linear algebra, we don't "do" algebra on the rank of a matrix and, on the other hand, that in multilinear algebra we are "doing" algebra on the rank of a k-tensor of dim(n^k)? This is what I'm aiming for when I ask whether the 2-tensors can also modify the rank of a matrix, in addition to all the elementary row operations.
Thank you for your consideration
Not really. Elementary tensors of the form $v \otimes w$ where $v = (v_1,\dots,v_n)^T \in \mathbb{R}^{n \times 1}$ is a column vector and $w = (w_1, \dots, w_m) \in \mathbb{R}^{1 \times m}$ is a row vector correspond to the $n \times m$ matrix
$$ v \cdot w = \begin{pmatrix} v_1 w_1 & v_1 w_2 & \dots & v_1 w_m \\ \vdots & \vdots & \ddots & \vdots \\ v_n w_1 & v_n w_2 & \dots & v_n w_m \end{pmatrix}. $$
This is a rank one matrix if $v \neq 0$ or $w \neq 0$ and it is definitely not an elementary matrix (which must be square and invertible). A general tensor $T \in \mathbb{R}^{n \times 1} \otimes \mathbb{R}^{1 \times m}$ is a linear combination of tensors of the form $v_i \otimes w_i$ such as
$$ T = a_1 \cdot (v_1 \otimes w_1) + \dots + a_l \cdot (v_l \otimes w_l) $$
and this corresponds to the matrix $A = a_1 (v_1 \cdot w_1) + \dots + a_l (v_l \cdot w_l)$.
Given a tensor $T$, we can ask what is the minimal number $k$ of elementary tensors such that $T$ can be represented as a linear combination of $k$ elementary tensors. This number is called the tensor rank of $k$ and under the correspondence between tensors and matrices this number is precisely the rank of the matrix $A = a_1 (v_1 \cdot w_1) + \dots + a_l (v_l \cdot w_l)$.