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I have an $R^{n \times n \times n \times n}$ tensor that maps a matrix to another matrix, call it $K$. I also have the matrix $C = A \times B$ where $C,A,B \in \mathbb{R}^{n \times n}$ and $\times$ is just matrix product.

Is there a way to calculate $A\times K(B)$ by applying $K$ somehow on $C$ or doing some other operations on $C$ to get $A \times K(B)$?

Thanks.

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If you must use tensor language to ask your question, can you please write whatever you're trying to ask in terms of tensor contraction? Your "$R^{n \times n \times n \times n}$ tensor" is a "rank 4 tensor on $R^n$", in more standard terminology. Setting $V = {\mathbf R}^n$, your $K$ belongs to $(V \otimes V^*)^{\otimes 2}$, and it maps $V \otimes V^*$ to itself by tensor contraction in several possible ways (there is more than one way to contract). –  KCd May 17 '12 at 14:53
    
@KCd, thanks for the comment. I am actually talking about $K$ being a very specific tensor, the one that appears in the first answer in math.stackexchange.com/questions/146052/… . –  kloop May 17 '12 at 15:19

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