Zero Matrix is Identity for Matrix Entrywise Addition/Proof 1

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

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).

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.

$\blacksquare$