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Matrix multiplication can be thought of as a matrix of the sum of the products of the matrix elements. That is

$\mathbf{AB}=\mathbf{C}$

where

$c_{ij} = \sum_{n=1}^\max a_{in}b_{nj}$

Is there a form of matrix multiplication where the result is the product of the matrix elements? That is

$c_{ij} = \prod_{n=1}^\max a_{in}b_{nj}$

If so, what is the terminology? Is it still called matrix multiplication? Is there a whole different term for it? And has there been much mathematics developed from this variant of multiplication?

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    $\begingroup$ Your first formula differs from usual matrix multiplication. $\endgroup$
    – coffeemath
    May 10, 2021 at 0:20
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    $\begingroup$ The usual matrix multiplication (which you seem to have transcribed incorrectly) arises when trying to understand matrices as representations of linear operators acting on a space. There is a geometric intuition behind this (think about how a matrix acts on basis vectors). The Kronecker product has applications in probability and, it seems, robotics. The Hadamard product (term by term multiplication) has uses in image processing. What application does this proposed multiplication of your have? $\endgroup$
    – Xander Henderson
    May 10, 2021 at 0:44
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    $\begingroup$ Keep in mind that the actual matrix multiplication didn't arise in a vacuum - it's what you get when you think about how to represent composition, given the interpretation of matrices as maps between vector spaces. Briefly: if $A$ and $B$ are the matrices representing linear maps $f$ and $g$ respectively, then $AB$ represents the map $g\circ f$ (to preempt a reasonable worry note that the product parses if and only if the composition makes sense; also, all of this assumes a choice of bases for the spaces involved). So it's not just some random choice. $\endgroup$ May 10, 2021 at 0:53
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    $\begingroup$ Note the proposed multiplication always gives something rank 1, a product of products can be rearranged. So we wont have identity or the like. $\endgroup$ May 10, 2021 at 0:58
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    $\begingroup$ One thing is for sure. If you say "matrix multiplication", I assure you people are not going to think about this operation, so my advice is don't call it that. You may say this is yet another of the infinitely many possible binary operations defined on the set $M_{m\times n}$ of $m\times n$ matrices. $\endgroup$ May 10, 2021 at 1:07

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Note that $c_{ij} = \displaystyle{\prod_{k=1}^n} a_{ik}b_{kj}=\left(\displaystyle{\prod_{k=1}^n} a_{ik}\right)\left(\displaystyle{\prod_{k=1}^n} b_{kj}\right)$.

For convenience, define a function $\mu$ that takes a matrix and returns the vector of the products of its rows, so that $\left(\mu(\mathbf A)\right)_i=\displaystyle{\prod_{k=1}^n} a_{ik}$ and, using $\top$ for transpose, $\left(\mu(\mathbf B^\top)\right)_j=\displaystyle{\prod_{k=1}^n} b_{kj}$. Then we have $c_{ij}=\left(\mu(\mathbf A)\right)_i\left(\mu(\mathbf B^\top)\right)_j$.

This means that the matrix $\mathbf C$ is the outer product $\mu(\mathbf A)\otimes \mu(\mathbf B^\top)$. If we write vectors like $\mu(\mathbf A)$ as column vectors, then we can write $\mathbf C$ using traditional matrix multiplication: $\mathbf C=\mu(\mathbf A)\mu(\mathbf B^\top)^\top$.

Because of the above calculation using the known operation of "outer product" and throwing away all information about $\mathbf A$ (resp. $\mathbf B$) except for the products of the rows (resp. columns), I doubt there is a name for this $\mathbf C$ sort of product. However, I would be curious if anyone has encountered a standard name/symbol for what I called "$\mu$" above.

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