# Matrix Entrywise Addition forms Abelian Group

## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$.

Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group.

## Proof

We have by definition that matrix entrywise addition is a specific instance of a Hadamard product.

By definition of a ring, the structure $\struct {R, +}$ is a group.

As $\struct {R, +}$ is a fortiori a monoid, it follows from Matrix Space Semigroup under Hadamard Product that $\struct {\map {\MM_R} {m, n}, +}$ is also a monoid.

As $\struct {R, +}$ is a group, it follows from Negative Matrix is Inverse for Matrix Entrywise Addition that all elements of $\struct {\map {\MM_R} {m, n}, +}$ have an inverse element.

From Matrix Entrywise Addition is Commutative it follows that $\struct {\map {\MM_R} {m, n}, +}$ is an Abelian group.

The result follows.

$\blacksquare$

## Examples

### $2 \times 2$ Matrices over Rational Numbers

Let $\Q^{2 \times 2}$ denote the set of order $2$ square matrices over the set $\Q$ of rational numbers.

Then the algebraic structure $\struct {\Q^{2 \times 2}, +}$, where $+$ denotes matrix entrywise addition, is an abelian group.

### $n \times n$ Matrices over Real Numbers

Let $\R^{n \times n}$ denote the set of order $n$ square matrices over the set $\R$ of real numbers.

Then the algebraic structure $\struct {\R^{n \times n}, +}$, where $+$ denotes matrix entrywise addition, is an abelian group.