# Definition:Matrix Entrywise Addition

## Definition

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

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

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.

## Examples

### Addition of Real $2 \times 2$ Matrices

Let $\mathbf A = \begin {pmatrix} p & q \\ r & s \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} w & x \\ y & z \end {pmatrix}$ be order $2$ square matrices over the real numbers.

Then the matrix sum of $\mathbf A$ and $\mathbf B$ is given by:

- $\mathbf A + \mathbf B = \begin {pmatrix} p + w & q + x \\ r + y & s + z \end {pmatrix}$

## Also see

There are several types of addition defined on matrices.

**Matrix entrywise addition**is the most common, and (at elementary level) it is just known as**matrix addition**

When more than one is being used during the course of an exposition, it is a *very good idea* to specify them with their full names whenever invoked.

- Results about
**matrix entrywise addition**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 29$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.2$: Some examples of rings: Ring Example $7$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**addition**(of matrices)