Elementary Matrix corresponding to Elementary Column Operation/Scale Column

Theorem

Let $\mathbf I$ denote the unit matrix of order $n$ over a field $K$.

Let $e$ be the elementary column operation acting on $\mathbf I$ as:

$(\text {ECO} 1)$

$:$

$\ds \kappa_k \to \lambda \kappa_k $

For some $\lambda \in K_{\ne 0}$, multiply column $k$ of $\mathbf I$ by $\lambda$

for $1 \le k \le n$.

Let $\mathbf E$ be the elementary column matrix defined as:

$\mathbf E = e \paren {\mathbf I}$

$\mathbf E$ is the square matrix of order $n$ of the form:

$E_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \lambda \cdot \delta_{a b} & : a = k \end{cases}$

where:

$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$

$\delta_{a b}$ is the Kronecker delta:

$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$

By definition of the unit matrix:

$I_{a b} = \delta_{a b}$

where:

$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.

By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.

That is, all elements of column $k$ of $\mathbf I$ are to be multiplied by $\lambda$.

By definition of unit matrix, all elements of column $k$ are $0$ except for element $I_{k k}$, which is $1$.

Thus in $\mathbf E$:

$E_{k k} = \lambda \cdot 1 = \lambda$

The elements in all the other columns of $\mathbf E$ are the same as the corresponding elements of $\mathbf I$.

Hence the result.

$\blacksquare$