Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns
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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:
\((\text {ECO} 3)\) | $:$ | \(\ds \kappa_i \leftrightarrow \kappa_j \) | Interchange columns $i$ and $j$ |
for $1 \le i \le n$, $1 \le j \le n$.
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} & : \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}$
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}$
Proof
By definition of the unit matrix:
- $I_{a b} = \delta_{a b}$
where:
By definition, $\mathbf E$ is the square matrix of order $n$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of column $i$ of $\mathbf I$ are to be exchanged with the corresponding elements of column $j$.
By definition of unit matrix:
- all elements of column $i$ are $0$ except for element $I_{i i}$, which is $1$.
- all elements of column $j$ are $0$ except for element $I_{j j}$, which is $1$.
Thus in $\mathbf E$:
- where $a \ne i$ and $a \ne j$, $E_{a b} = \delta_{a b}$ (all columns except $i$ and $j$ are unchanged)
- where $a = i$, $E_{a b} = \delta_{a j}$ (the contents of column $j$)
- where $a = j$, $E_{a b} = \delta_{a i}$ (the contents of column $i$)
That is:
- $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}$
Hence the result.
$\blacksquare$