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 $\mathbf B$.

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.