# How do we describe standard matrix multiplication using tensor products?

Let $V$ be a finite dimensional vector space over a field $F$. Consider the bilinear map $End(V) \times End(V) \rightarrow End(V)$ given by $(u,v) \rightarrow u \circ v$ and the map associated linear map of tensor productsw $m : End(V) \otimes End(V) \rightarrow End(V)$.

I am interested in how we can identify an element of the tensor product $End(V)^* \otimes End(V)^* \otimes End(V)$ with the map $m$ which apperently is just another way to describe standard multiplication of matrices. In any case the question can be formulated as follows:

How do we identify $m$ with an element $u^* \otimes v^* \otimes w \in End(V)^* \otimes End(V)^* \otimes End(V)$?

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I believe it will be a sum of tensors, not a simple tensor. Choosing a basis of $V$, there is an element $u^* = a_{ij}$ that takes a matrix $A$ and gives you its $i,j$th entry. Set $b_{ij} = a_{ij}$ to be a nicer name for $v^*$, and $e_{ij}$ to be the matrix unit with a 1 in the $i,j$th spot, and 0 elsewhere. Then the matrix multiplication element is: $$\mu = \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n a_{ij} \otimes b_{jk} \otimes e_{ik}$$ In other words, it is just the formula for matrix multiplication with some tensor products instead of multiplication signs.
Depending on taste, one might also have the "motion" go in the opposite direction: with $v\otimes \lambda$ an endomorphism given by $(v\otimes \lambda)(w)=\lambda(w)\cdot v$, for $v,w\in V$ and $\lambda\in V^*$, the multiplication/composition of endomorphisms is $$(v\otimes \lambda)\circ (w\otimes \mu) \;=\; \lambda(w)\cdot v\otimes \mu$$ for $v,w\in V$ and $\lambda,\mu\in V^*$.