Definition:Matrix Entrywise Addition

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

Let the dimensions 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.

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


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