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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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up vote 3 down vote accepted

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.

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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 $\lambda,\mu\in V^*$.

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