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According to Wikipedia's page on tensor contraction:

In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V^* \otimes \ldots \otimes V^*$ (where there are $m$ $V$ factors and $n$ $V^*$ factors). Applying the natural pairing to the $k$th $V$ factor and the $l$th $V^*$ factor, and using the identity on all other factors, defines the $(k,l)$ contraction operation, which is a linear map which yields a tensor of type $(m-1, n-1)$.

I must admit, I'm having trouble understanding this definition. What is the actual map? From what I gather, the $(k,l)$ contraction of, say, $$T = X_1 \otimes \ldots \otimes X_m \otimes \omega^1 \otimes \ldots \otimes \omega^n$$ is $$C(T) = \omega^l(X_k)\cdot X_1 \otimes \ldots \otimes \widehat{X_k} \otimes \ldots \otimes X_m \otimes \omega^1 \otimes \ldots \otimes \widehat{\omega^l} \otimes \ldots \otimes \omega^n,$$ where the hats indicate omission, and $C(T)$ is just my notation for contraction.

Is this correct? If so, is this often taken as the definition or are there other standard (equivalent) definitions?

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    Relevant: http://math.stackexchange.com/questions/36042/trace-of-a-bilinear-form2011-06-26

2 Answers 2