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Assume $A,X,Y\in \mathbb{R}^{I_1\times\cdots\times I_N}$ are three arbitrary N-th order tensors.

How to prove the following equation: $$\langle A,X\rangle Y=(X\otimes Y)(A),$$ where $\langle\cdot ,\cdot\rangle$ means the inner product of tensors, i.e. $\langle A,X\rangle=\sum\limits_{i_1=1}^{I_1}\sum\limits_{i_2=1}^{I_2}\cdots\sum\limits_{i_N=1}^{I_N}a_{i_1i_2\cdots i_N} x_{i_1i_2\cdots i_N}$, and actually here the product $(X\otimes Y)$ is an operator from $\mathbb{R}^{I_1\times\cdots\times I_N}$ to itself $\mathbb{R}^{I_1\times\cdots\times I_N}$.

Thank you very much.

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  • $\begingroup$ I'd have said that this was the definition of $(X\otimes Y)$. What definition have you been given? $\endgroup$ Feb 3 '16 at 15:21
  • $\begingroup$ Thanks for your comment. I'm not familiar with tensor analysis. Could you please give some reference on this definition? $\endgroup$
    – peter
    Feb 3 '16 at 17:10
  • $\begingroup$ This is the outer product of $X$ and $Y$. $\endgroup$ Feb 3 '16 at 17:13

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