Hadamard Product over Group forms Group

Theorem

Let $\struct {G, \cdot}$ be a group whose identity is $e$.

Let $\map {\MM_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \cdot}$.

Then $\struct {\map {\MM_G} {m, n}, \circ}$, where $\circ$ is Hadamard product, is also a group.

Proof

As $\struct {G, \cdot}$, being a group, is a monoid, it follows from Matrix Space Semigroup under Hadamard Product that $\struct {\map {\MM_G} {m, n}, \circ}$ is also a monoid.

As $\struct {G, \cdot}$ is a group, it follows from Negative Matrix is Inverse for Hadamard Product that all elements of $\struct {\map {\MM_G} {m, n}, \circ}$ have an inverse.

The result follows.

$\blacksquare$

Also see

• Matrix Entrywise Addition forms Abelian Group