Definition:Matrix Entrywise Addition

Definition
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {S, \circ}$.

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

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

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

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

Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the operation $\circ$ on corresponding entries of $\mathbf A$ and $\mathbf B$.

This operation is called matrix entrywise addition (or just matrix addition).

It follows that matrix entrywise addition is defined only when both matrices have the same number of rows and the same number of columns.

Also see

 * Definition:Matrix Addition, where can be found different operations on matrices also referred to as addition:
 * Definition:Matrix Direct Sum
 * Definition:Kronecker Sum