Matrix Entrywise Addition is Associative/Proof 1
Theorem
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is associative on $\map \MM {m, n}$.
That is:
- $\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$
for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map \MM {m, n}$.
Proof
From:
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.
Hence we can apply Matrix Entrywise Addition over Ring is Associative.
$\Box$
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.
By definition, matrix entrywise addition is the Hadamard product with respect to addition of numbers.
The result follows from Associativity of Hadamard Product.
$\blacksquare$