Properties of Hadamard Product

Let $$\mathcal {M}_{S} \left({m, n}\right)$$ be a $m \times n$ matrix space over $$S$$ over an algebraic structure $$\left({S, \circ}\right)$$.

Let $$\mathbf{A}, \mathbf{B} \in \mathcal {M}_{S} \left({m, n}\right)$$.

Let $$\mathbf{A} + \mathbf{B}$$ be defined as the matrix entrywise sum of $$\mathbf{A}$$ and $$\mathbf{B}$$.

The operation of matrix entrywise addition satisfies the following properties.


 * $$+$$ is closed on $$\mathcal {M}_{S} \left({m, n}\right)$$ iff $$\circ$$ is closed on $$\left({S, \circ}\right)$$.
 * $$+$$ is associative on $$\mathcal {M}_{S} \left({m, n}\right)$$ iff $$\circ$$ is associative on $$\left({S, \circ}\right)$$.
 * $$+$$ is commutative on $$\mathcal {M}_{S} \left({m, n}\right)$$ iff $$\circ$$ is commutative on $$\left({S, \circ}\right)$$.

Closure

 * Let $$\left[{a}\right]_{m n}, \left[{b}\right]_{m n}$$ be elements of $$\mathcal {M}_{S} \left({m, n}\right)$$.

Let $$\left[{c}\right]_{m n} = \left[{a}\right]_{m n} + \left[{b}\right]_{m n}$$.

Then $$\forall i \in \left[{1 \,. \, . \, m}\right], j \in \left[{1 \,. \, . \, n}\right]: c_{i j} = a_{i j} \circ b_{i j}$$.

Thus $$\left({S, \circ}\right)$$ is closed iff $$c_{i j} \in S$$.

As $$\left[{c}\right]_{m n}$$, from the definition of matrix addition, has the same dimensions as both $$\left[{a}\right]_{m n}$$ and $$\left[{b}\right]_{m n}$$, it follows that $$\left[{c}\right]_{m n} \in \mathcal {M}_{S} \left({m, n}\right)$$.

Thus $$\left({\mathcal {M}_{S} \left({m, n}\right), +}\right)$$, as it is defined, is closed.

Hence the result.

The argument reverses.

Associativity
Let $$\left[{a}\right]_{m n}, \left[{b}\right]_{m n}, \left[{c}\right]_{m n}$$ be elements of $$\mathcal {M}_{S} \left({m, n}\right)$$.

Then let:


 * $$\left[{p}\right]_{m n} = \left({\left[{a}\right]_{m n} + \left[{b}\right]_{m n}}\right) + \left[{c}\right]_{m n}$$;
 * $$\left[{q}\right]_{m n} = \left[{a}\right]_{m n} + \left({\left[{b}\right]_{m n} + \left[{c}\right]_{m n}}\right)$$.

Let $$\circ$$ be associative on $$\left({S, \circ}\right)$$.

$$ $$ $$


 * Now let matrix addition on $$\mathcal {M}_{S} \left({m, n}\right)$$ be associative.

Then it follows trivially that $$\circ$$ is associative on $$\left({S, \circ}\right)$$.

Commutativity
Let $$\left[{a}\right]_{m n}, \left[{b}\right]_{m n}$$ be elements of $$\mathcal {M}_{S} \left({m, n}\right)$$.

Let:
 * $$\left[{c}\right]_{m n} = \left[{a}\right]_{m n} + \left[{b}\right]_{m n}$$.
 * $$\left[{d}\right]_{m n} = \left[{b}\right]_{m n} + \left[{a}\right]_{m n}$$.

Let $$\circ$$ be commutative on $$\left({S, \circ}\right)$$.

$$ $$ $$


 * Now let matrix addition on $$\mathcal {M}_{S} \left({m, n}\right)$$ be commutative.

Then it follows trivially that $$\circ$$ is commutative on $$\left({S, \circ}\right)$$.