# Zero Matrix is Identity for Matrix Entrywise Addition

## Theorem

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.

Then $\mathbf 0$ is the identity element for matrix entrywise addition.

## Proof 1

From:

the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings whose zero is the number $0$ (zero).

Hence we can apply Zero Matrix is Identity for Matrix Entrywise Addition over Ring.

$\Box$

The above cannot be applied to the natural numbers $\N$, as they do not form a ring.

However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid whose identity is $0$ (zero).

By definition, matrix entrywise addition is the **Hadamard product** with respect to addition of numbers.

The result follows from Zero Matrix is Identity for Hadamard Product.

$\blacksquare$

## Proof 2

Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.

Then:

\(\ds \mathbf A + \mathbf 0\) | \(=\) | \(\ds \sqbrk a_{m n} + \sqbrk 0_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \sqbrk {a + 0}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||

\(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Identity Element of Addition on Numbers | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \mathbf A + \mathbf 0\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix |

Similarly:

\(\ds \mathbf 0 + \mathbf A\) | \(=\) | \(\ds \sqbrk 0_{m n} + \sqbrk a_{m n}\) | Definition of $\mathbf A$ and $\mathbf 0_R$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \sqbrk {0 + a}_{m n}\) | Definition of Matrix Entrywise Addition | |||||||||||

\(\ds \) | \(=\) | \(\ds \sqbrk a_{m n}\) | Identity Element of Addition on Numbers | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \mathbf 0 + \mathbf A\) | \(=\) | \(\ds \mathbf A\) | Definition of Zero Matrix |

$\blacksquare$

## Also see

## Sources

- 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices: $3$