Is there an alternative matrix multiplication? 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?
 A: 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.
