Definition:Matrix Entrywise Addition/Motivation

Matrix Entrywise Addition: Motivation
Consider the linear transformations:

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:

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.