Properties of Hadamard Product

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

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

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

The operation of Hadamard product satisfies the following properties:


 * $+$ is closed on $\map {\mathcal M_S} {m, n}$ $\circ$ is closed on $\struct {S, \circ}$
 * $+$ is associative on $\map {\mathcal M_S} {m, n}$ $\circ$ is associative on $\struct {S, \circ}$
 * $+$ is commutative on $\map {\mathcal M_S} {m, n}$ $\circ$ is commutative on $\struct {S, \circ}$.