I know this might be somewhat of a dumb question, but I've discovered a neat way to represent the components of a (assume real valued) tensor via a sort of "multiplication". Let's say I have a real $(0,2)$ tensor (order $2$) of dimension $2$: $$\mathbf{T} =\begin{bmatrix} \begin{bmatrix} T_{1,1} & T_{1,2} \end{bmatrix} & \begin{bmatrix} T_{2,1} & T_{2,2} \end{bmatrix} \end{bmatrix}$$ Since this is a $(0,2)$ tensor, I right "multiply" by $2$ contravariant basis vectors to retrieve components. For example if I want the component $T_{1,2}$ I take the right product with $\underline{e_1}$ and $\underline{e_2}$: $$\begin{bmatrix} \begin{bmatrix} T_{1,1} & T_{1,2} \end{bmatrix} & \begin{bmatrix} T_{2,1} & T_{2,2} \end{bmatrix} \end{bmatrix}\begin{bmatrix} 1\\ 0 \end{bmatrix}\begin{bmatrix} 0\\ 1 \end{bmatrix}=\begin{bmatrix} T_{1,1} & T_{1,2} \end{bmatrix}\begin{bmatrix} 0\\ 1 \end{bmatrix}=T_{1,2}$$ Which follows the usual standards of "matrix" multiplication. Another example: Supposing we have a $(2,1)$ tensor (order $3$) of dimension $2$: $$\mathbf{T} =\begin{bmatrix} \begin{bmatrix} T^{1,1}_{1}\\ T^{1,2}_{1} \end{bmatrix} & \begin{bmatrix} T^{1,1}_{2}\\ T^{1,2}_{2} \end{bmatrix}\\ \begin{bmatrix} T^{2,1}_{1}\\ T^{2,2}_{1} \end{bmatrix} & \begin{bmatrix} T^{2,1}_{2}\\ T^{2,2}_{2} \end{bmatrix} \end{bmatrix}$$ Since this is a $(2,1)$ tensor I left multiply by two covariant basis vectors and right multiply by one contravariant basis vector. Say if I want to retrieve $T^{2,1}_2$ - Then I can left multiply by $\underline{\epsilon^2}$ and $\underline{\epsilon^1}$ (the dual space basis) and right multiply by $\underline{e_2}$ (Notice that the order we write the covectors is opposite to their index order so that the order of multiplication matches the index order) $$\begin{bmatrix} 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \end{bmatrix}\begin{bmatrix} \begin{bmatrix} T^{1,1}_{1}\\ T^{1,2}_{1} \end{bmatrix} & \begin{bmatrix} T^{1,1}_{2}\\ T^{1,2}_{2} \end{bmatrix}\\ \begin{bmatrix} T^{2,1}_{1}\\ T^{2,2}_{1} \end{bmatrix} & \begin{bmatrix} T^{2,1}_{2}\\ T^{2,2}_{2} \end{bmatrix} \end{bmatrix}\begin{bmatrix} 0\\ 1 \end{bmatrix}$$
$$\begin{bmatrix} 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \end{bmatrix}\begin{bmatrix} \begin{bmatrix} T^{1,1}_{2}\\ T^{1,2}_{2} \end{bmatrix}\\ \begin{bmatrix} T^{2,1}_{2}\\ T^{2,2}_{2} \end{bmatrix} \end{bmatrix}$$
$$=\begin{bmatrix} 1 & 0 \end{bmatrix}\begin{bmatrix} T^{2,1}_{2}\\ T^{2,2}_{2} \end{bmatrix}=T^{2,1}_{2}$$ My question is: is there a name for this operation? It really resembles matrix multiplication but isn't, since the item in the middle isn't a matrix - it is a tensor that I have simply "projected down" so it is readable on a 2D screen. Is there any special notation for this operation?
Thanks for your time,
-K. defaoite