Definition:Matrix Entrywise Addition/Ring

Definition
Let $\struct {R, +, \cdot}$ be a ring.

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

Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$.

Then the matrix entrywise sum of $\mathbf A$ and $\mathbf B$ is written $\mathbf A + \mathbf B$, and is defined as follows.

Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$.

Then:
 * $\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$

Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the (ring) addition operation $+$ on corresponding entries of $\mathbf A$ and $\mathbf B$. That is, the matrix entrywise sum of $\mathbf A$ and $\mathbf B$ is the Hadamard product of $\mathbf A$ and $\mathbf B$ with respect to ring addition.

This operation is called matrix entrywise addition.

Also defined as
Some sources restrict their attention to this operation to such matrices whose underlying structures are fields.