Definition:Matrix Entrywise Addition/Motivation

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Matrix Entrywise Addition: 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.


Sources

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