Elementary Matrix corresponding to Elementary Column Operation

Theorem

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

Let $e$ be an elementary column operation on $\mathbf I$.

Let $\mathbf E$ be the elementary column matrix of order $n$ uniquely defined as:

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

where $\mathbf I$ is the unit matrix.

Let $\kappa_k$ denote the $k$th column of $\mathbf I$ for $1 \le k \le n$.

Case $(1)$: Scalar Product of Column

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 $\lambda$

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

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

Case $(2)$: Add Scalar Product of Column to Another

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

\((\text {ECO} 2)\)

$:$

\(\ds \kappa_i \to \kappa_i + \lambda r_j \)

For some $\lambda \in K$, add $\lambda$ times column $j$ to row $i$

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

$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{b i} \cdot \delta_{j a}$

Case $(3)$: Exchange Columns

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

\((\text {ECO} 3)\)

$:$

\(\ds \kappa_i \leftrightarrow \kappa_j \)

Interchange columns $i$ and $j$

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

$E_{a b} = \begin {cases} \delta_{a b} & : \text {if $b \ne i$ and $b \ne j$} \\ \delta_{a j} & : \text {if $b = i$} \\ \delta_{a i} & : \text {if $b = j$} \end {cases}$

Throughout the above:

$E_{a b}$ 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}$

Also see

• Elementary Matrix corresponding to Elementary Row Operation