Hadamard Product of Matricies

Let $A,B:[m]\times[n]\to M$ be a pair of Matrices over a Monoid. The Hadamard Product $A\odot B:[m]\times[n]\to M$ is the matrix formed by taking the elementwise product of $A$ and $B$.

Properties

The Hadamard product is associative and unital with respect to the matrix containing only $\varepsion$.

If $M$ is commutative, then the Hadamard product is also commutative. If the two matrices are taken over a Ring $R$, then the Hadamard product is distributive with respect to Addition of Matrices.