Definition:Matrix Entrywise Addition/Ring

From ProofWiki
Jump to navigation Jump to search

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 $\mathbf B$.

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 ring addition.


This operation is called matrix entrywise addition.


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.


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.


Also defined as

Some sources restrict their attention to this operation to such matrices whose underlying structures are fields.


Also see

  • Results about matrix entrywise addition can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Matrix_Entrywise_Addition/Ring&oldid=540070"