Definition:Matrix Entrywise Addition

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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

  • Results about matrix entrywise addition can be found here.


Sources

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