Definition:Matrix Entrywise Addition

Definition
Let $$\mathcal M_S \left({m, n}\right)$$ be a $m \times n$ matrix space over $$S$$ over an algebraic structure $$\left({S, \circ}\right)$$.

Let $$\mathbf A, \mathbf B \in \mathcal M_S \left({m, n}\right)$$.

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 $$\mathbf A + \mathbf B = \mathbf C = \left[{c}\right]_{m n}$$.

Then $$\forall i \in \left[{1 \,. \, . \, m}\right], j \in \left[{1 \,. \, . \, n}\right]: c_{i j} = a_{i j} \circ b_{i j}$$.

Thus $$\left[{c}\right]_{m n}$$ is the $$m \times n$$ matrix whose elements are made by performing the operation $$\circ$$ on corresponding elements 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.

Note
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;


 * Matrix direct sum;


 * Kronecker sum.

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.