# 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:

Integers form Ring
Rational Numbers form Ring
Real Numbers form Ring
Complex Numbers form Ring

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).

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

$\blacksquare$