Properties of Hadamard Product

Theorem
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \cdot}$.

Let $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$.

Let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$.

The operation $\circ$ of Hadamard product satisfies the following properties:


 * $\circ$ is closed on $\map {\MM_S} {m, n}$ $\cdot$ is closed on $\struct {S, \cdot}$
 * $\circ$ is associative on $\map {\MM_S} {m, n}$ $\cdot$ is associative on $\struct {S, \cdot}$
 * $\circ$ is commutative on $\map {\MM_S} {m, n}$ $\cdot$ is commutative on $\struct {S, \cdot}$.