# Definition:Matrix Entrywise Addition

## Definition

Let $\mathbf A$ and $\mathbf B$ be matrices of numbers.

Let the orders of $\mathbf A$ and $\mathbf B$ both be $m \times n$.

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 $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding 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 addition of numbers.

This operation is called **matrix entrywise addition**.

### General Ring

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 $\mathbf B$.

### Defined Operation

It needs to be noted that the operation of **Hadamard product** is defined only when both matrices have the same number of rows and the same number of columns.

This restriction applies to the operation of **matrix entrywise addition**, which can be considered as a specific application of the **Hadamard product**.

### General Operation

Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$ be matrices all of order of $m \times n$.

Then the **matrix entrywise sum of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$** is written $\mathbf A_1 + \mathbf A_2 + \ldots + \mathbf A_k$, and is defined as follows:

Then:

- $\forall i \in \closedint 1 m, j \in \closedint 1 n: K_{i j} = \paren {a_1}_{i j} + \paren {a_2}_{i j} + \cdots + \paren {a_k}_{i j}$

where $\paren {a_l}_{i j}$ is the element of $\mathbf A_l$ whose indices are $\tuple {i, j}$.

Thus $\mathbf K = \sqbrk k_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$.

This can be expressed in summation form as:

- $\ds K = \sum_{j \mathop = 1}^k \mathbf A_j$

## 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:

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

### Arbitrary Matrices

Let $\mathbf A = \begin {pmatrix} 3 & -4 & 2 \\ -1 & 0 & 5 \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} 2 & 6 & -1 \\ 4 & -3 & 2 \end {pmatrix}$ be order $2$ square matrices over the real numbers.

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

- $\mathbf A + \mathbf B = \begin {pmatrix} 5 & 2 & 1 \\ 3 & -3 & 7 \end {pmatrix}$

## Also known as

The operation of **matrix entrywise addition** is usually referred to as just **matrix addition**.

Similarly, the **matrix entrywise sum** is likewise usually just called the **matrix sum**.

However, it needs to be made clear that there are a number of different operations on matrices which are also referred to as **matrix addition**, so there are contexts in which it is wise to make clear which is meant.

## Motivation

Consider the linear transformations:

\(\text {(1)}: \quad\) | \(\ds y_1\) | \(=\) | \(\ds \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n\) | |||||||||||

\(\ds y_2\) | \(=\) | \(\ds \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n\) | ||||||||||||

\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||

\(\ds y_m\) | \(=\) | \(\ds \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n\) |

\(\text {(2)}: \quad\) | \(\ds z_1\) | \(=\) | \(\ds \beta_{1 1} x_1 + \beta_{1 2} x_2 + \cdots + \beta_{1 n} x_n\) | |||||||||||

\(\ds z_2\) | \(=\) | \(\ds \beta_{2 1} x_1 + \beta_{2 2} x_2 + \cdots + \beta_{2 n} x_n\) | ||||||||||||

\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||

\(\ds z_m\) | \(=\) | \(\ds \beta_{m 1} x_1 + \beta_{m 2} x_2 + \cdots + \beta_{m n} x_n\) |

Let a new set of variables $w_1, w_2, \ldots, w_m$ be introduced by adding the corresponding $y_i$ and $z_i$:

$w_1 = y_1 + z_1, w_2 = y_2 + z_2, \ldots, w_m = y_m + z_m$

Then we have immediately:

\(\text {(3)}: \quad\) | \(\ds w_1\) | \(=\) | \(\ds \paren {\alpha_{1 1} + \beta_{1 1} } x_1 + \paren {\alpha_{1 2} + \beta_{1 2} } x_2 + \cdots + \paren {\alpha_{1 n} + \beta_{1 n} } x_n\) | |||||||||||

\(\ds w_2\) | \(=\) | \(\ds \paren {\alpha_{2 1} + \beta_{2 1} } x_1 + \paren {\alpha_{2 2} + \beta_{2 2} } x_2 + \cdots + \paren {\alpha_{2 n} + \beta_{2 n} } x_n\) | ||||||||||||

\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||

\(\ds w_m\) | \(=\) | \(\ds \paren {\alpha_{m 1} + \beta_{m 1} } x_1 + \paren {\alpha_{m 2} + \beta_{m 2} } x_2 + \cdots + \paren {\alpha_{m 1} + \beta_{m n} } x_n\) |

We can obtain $(3)$ from $(1)$ and $(2)$ by **adding** the linear transformations.

Hence when we express $(1)$ and $(2)$ by means of matrices $\mathbf A = \sqbrk \alpha_{m n}$ and $\mathbf B = \sqbrk \beta_{m n}$, the concept of **matrix entrywise addition** evolves naturally.

## Also see

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

- Definition:Hadamard Product: the same operation induced by a binary operation of a general algebraic structure

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

## Sources

- 1954: A.C. Aitken:
*Determinants and Matrices*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $5$. The Operations of Matrix Algebra - 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**addition**(of matrices)