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Matrix multiplication can be defined as:

$$({\bf A}{\bf B})_{kl} = \sum_{i=1}^{n} {\bf A}_{ki}{\bf B}_{il}$$

Lately, I have been thinking of another operation:

$$({\bf A} \vec{\circ} {\bf B})_{kl} = \prod_{i=1}^{n} {{\bf A}_{ki}}^{{\bf B}_{il}}$$

The ordinary one can be viewed as a linear weighting. The second one can be viewed as a geometric weighting. If working on non-negative reals this could for instance be used calculating probabilities. If we work with $\bf A$ over the complex numbers this is useful at least for averaging on a circle as can be done in my answer to this question. Also

$$\log({\bf A} \vec{\circ} {\bf A}^T)_{kl} = \sum_{i=1}^n {\bf A}_{ki} \log({\bf A}_{ki})$$ Which is the Shannon information entropy with reversed sign.

Does anyone know if this operation has a specific name? Can it be used for other things as well?

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mathreadler
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    One should avoid denoting this operation by ${\bf A} \otimes {\bf B}$, which is the standard notation for the Kronecker product of two matrices. https://en.wikipedia.org/wiki/Kronecker_product – Travis Willse Jul 09 '15 at 10:05
  • Yes sorry, I am actually aware of this. Will try and find some better un-used symbol. – mathreadler Jul 09 '15 at 10:07
  • What is $n$ in the indices? – GFauxPas Jul 09 '15 at 11:36
  • It is the size of the dimension. So that one full row of $\bf A$ and one full column of $\bf B$ is involved in each result. Nice trick by the way. – mathreadler Jul 09 '15 at 11:38

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