nyuuzyou/wikis huggingface dataset:
Element to Power of Group Order is Identity

== Theorem ==
Let $G$ be a group whose identity is $e$ and whose order is $n$.
Then:
:$\forall g \in G: g^n = e$
== Proof ==
Let $G$ be a group such that $\order G = n$.
Let $g \in G$ and let $\order g = k$.
From Order of Element Divides Order of Finite Group:
:$k \divides n$
So:
:$\exists m \in \Z_{>0}: k m = n$
Thus:
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.7$
* : $\S 1.9$: Theorem $12$ Corollary
* : $0$: Some Conventions and some Basic Facts
* : Chapter $7$: Cosets and Lagrange's Theorem: Exercise $16 \ \text{(i)}$
Category:Group Theory
Element to Power of Multiple of Order is Identity

== Theorem ==
Let $G$ be a group whose identity is $e$.
Let $a \in G$ have finite order such that $\order a = k$.
Then:
:$\forall n \in \Z: k \divides n \iff a^n = e$
where $k \divides n$ denotes that $k$ is a divisor of $n$.
== Proof ==
Let $k \in \N$ be the smallest such that $a^k = e$ as per the hypothesis.
=== Necessary Condition ===
Let $a^n = e$.
Let $n = q k + r, 0 \le r < k$.
By Element to Power of Remainder:
: $a^r = a^n = e$
But $0 \le r < k$.
Since $k$ is the smallest such that $a^k = e$:
:$1 \le s < k \implies a^s \ne e$
Thus $r = 0$.
That is:
: $k \divides n$
=== Sufficient Condition ===
Suppose $k \divides n$.
Then:
: $\exists s \in \Z: n = s k$
So:
== Sources ==
* : Chapter $\text {I}$: The Group Concept: $\S 8$: The Order (Period) of an Element: Theorem $3$
* : $\S 5.4$. Cyclic groups: Example $101$
* : $\S 1.4$: Theorem $3 \ (2)$
* : $0$: Some Conventions and some Basic Facts
* : $\S 38.4$ Period of an element: $\text{(ii)}$
Category:Identity Elements
Category:Order of Group Elements
Element to Power of Positive Characteristic of Field is Zero

== Theorem ==
Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.
Let the characteristic of $F$ be $n$ such that $n > 0$.
Then:
:$n \cdot a = 0$
where $n \cdot a$ denotes the power of $a$ in the context of the additive group $\struct {F, +}$:
:$n \cdot a = \begin {cases}
0 & : n = 0 \\
\paren {\paren {n - 1} \cdot a} + a & : n > 0
\end {cases}$
== Proof ==
By definition, the characteristic of $\struct {F, +, \times}$ is the order of the additive group $\struct {F, +}$.
By Element to Power of Group Order is Identity it follows directly that:
:$n \cdot a = 0$
== Sources ==
* : Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 89 \alpha$
Category:Characteristics of Fields
Element to Power of Remainder

== Theorem ==
Let $G$ be a group whose identity is $e$.
Let $a \in G$ have finite order such that $\order a = k$.
Then:
:$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$
== Proof ==
Let $n \in \Z$.
We have:
The result follows from Equal Powers of Finite Order Element.
== Sources ==
* : Chapter $2$: Subgroups and Cosets: $\S 41$
* : $\S 38.4$ Period of an element: $\text{(i)}$ 
Category:Group Theory
Element under Left Operation is Right Identity

== Theorem ==
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.
Then all of the elements of $\struct {S, \gets}$ are right identities.
== Proof ==
From Structure under Left Operation is Semigroup, $\struct {S, \gets}$ is a semigroup.
From the definition of left operation:
:$\forall x, y \in S: x \gets y = x$
from which it is apparent that all elements of $S$ are right identities.
== Also see ==
* Element under Right Operation is Left Identity
* All Elements of Left Operation are Left Zeroes
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$
Category:Left Operation
Category:Examples of Identity Elements
Element under Right Operation is Left Identity

== Theorem ==
Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.
Then all of the elements of $\struct {S, \to}$ left identities.
== Proof ==
From Structure under Left Operation is Semigroup, $\struct {S, \to}$ is a semigroup.
From the definition of right operation:
:$\forall x, y \in S: x \to y = y$
from which it is apparent that all elements of $S$ are left identities.
== Also see ==
* Element under Left Operation is Right Identity
* All Elements of Right Operation are Right Zeroes
== Sources ==
* : Chapter $4$. Groups: Exercise $4$
* : Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$
* : Chapter $5$: Semigroups: Exercise $3 \ \text{(i)}$
Category:Right Operation
Category:Examples of Identity Elements
Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse

== Theorem ==
Let $e$ be an elementary column operation.
Let $\mathbf E$ be the elementary column matrix corresponding to $e$.
Let $e'$ be the inverse of $e$.
Then the elementary column matrix corresponding to $e'$ is the inverse of $\mathbf E$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
Category:Elementary Column Operations
Category:Elementary Matrices
Category:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse
Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 1

== Theorem ==
== Proof ==
We will demonstrate this for each of the $3$ types of elementary column operation.
Let $\sim$ denote column equivalence.
Let $\mathbf I$ denote the unit matrix.
=== $\text {ECO} 1$: Scalar Product of Column ===
Let $e_1$ be the elementary column operation:
:$e_1 := \kappa_i \to \lambda \kappa_i$
for an arbitrary column $\kappa_i$ of $\mathbf I$.
Let $\mathbf E_1$ be the elementary column matrix created by applying $e_1$ to $\mathbf I$.
From Existence of Inverse Elementary Column Operation: Scalar Product of Column, the inverse $e_1'$ of $e_1$ is given by:
:$e_1' := \kappa_i \to \dfrac 1 \lambda \kappa_i$
Thus applying $e_1'$ to $\mathbf E_1$ transforms $\mathbf E_1$ back to $\mathbf I$.
From Elementary Column Operations as Matrix Multiplications, for every elementary column operation there exists a corresponding elementary column matrix. 
Let $\mathbf E_1'$ be the elementary column matrix created by applying $e_1'$ to $\mathbf I$.
Then:
:$\mathbf E_1 \mathbf E_1' = \mathbf E_1' \mathbf E_1 = \mathbf I$
=== $\text {ECO} 2$: Add Scalar Product of Column to Another ===
Let $e_2$ be the elementary column operation:
:$e_2 := \kappa_i \to \kappa_i + \lambda \kappa_j$
for arbitrary column $\kappa_i$ and $\kappa_j$ of $\mathbf I$ such that $i \ne j$.
Let $\mathbf E_2$ be the elementary column matrix created by applying $e_2$ to $\mathbf I$.
From Existence of Inverse Elementary Column Operation: Add Scalar Product of Column to Another, the inverse $e_2'$ of $e_2$ is given by:
:$e_2' := \kappa_i \to \kappa_i - \lambda r_j$
Thus applying $e_2'$ to $\mathbf E_2$ transforms $\mathbf E_2$ back to $\mathbf I$.
From Elementary Column Operations as Matrix Multiplications, for every elementary column operation there exists a corresponding elementary column matrix. 
Let $\mathbf E_2'$ be the elementary column matrix created by applying $e_2'$ to $\mathbf I$.
Then:
:$\mathbf E_2 \mathbf E_2' = \mathbf E_2' \mathbf E_2 = \mathbf I$
=== $\text {ECO} 3$: Exchange Columns ===
Let $e_3$ be the elementary column operation:
:$e_3 := \kappa_i \leftrightarrow \kappa_j$
for arbitrary column $\kappa_i$ and $\kappa_j$ of $\mathbf I$ such that $i \ne j$.
Let $\mathbf E_3$ be the elementary column matrix created by applying $e_3$ to $\mathbf I$.
From Existence of Inverse Elementary Column Operation: Exchange Columns, the inverse $e_3'$ of $e_3$ is $e_3$ itself:
:$e_3' := \kappa_i \leftrightarrow \kappa_j = e_3$
Thus applying $e_3$ to $\mathbf E_3$ transforms $\mathbf E_3$ back to $\mathbf I$.
:$\mathbf E_3 \mathbf E_3 = \mathbf I$
Hence the result, from Proof by Cases.
Category:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse
Elementary Column Matrix is Nonsingular

== Theorem ==
Let $\mathbf E$ be an elementary column matrix.
Then $\mathbf E$ is nonsingular.
== Proof ==
From Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse it is demonstrated that:
:if $\mathbf E$ is the elementary column matrix corresponding to an elementary column operation $e$
then:
:the inverse of $e$ corresponds to an elementary column matrix which is the inverse of $\mathbf E$.
So as $\mathbf E$ has an inverse, it is nonsingular.
== Also see ==
* Elementary Row Matrix is Nonsingular
Category:Elementary Matrices
Category:Nonsingular Matrices
Elementary Column Operations as Matrix Multiplications

== Theorem ==
Let $e$ be an elementary column operation.
Let $\mathbf E$ be the elementary column matrix of order $n$ defined as:
:$\mathbf E = e \paren {\mathbf I}$
where $\mathbf I$ is the unit matrix.
Then for every $m \times n$ matrix $\mathbf A$:
:$e \paren {\mathbf A} = \mathbf A \mathbf E$
where $\mathbf A \mathbf E$ denotes the conventional matrix product.
=== Corollary ===
== Proof ==
Let $s, t \in \closedint 1 m$ such that $s \ne t$.
=== Case $1$ ===
Let $e$ be the elementary column operation $\kappa_s \to \lambda \kappa_s$:
:$E_{k i} = \begin{cases}
\delta_{k i} & : i \ne s \\
\lambda \delta_{k i} & : i = s
\end{cases}$
where $\delta$ denotes the Kronecker delta.
Then:
=== Case $2$ ===
Let $e$ be the elementary column operation $\kappa_s \to \kappa_s + \lambda \kappa_t$:
:$E_{k i} = \begin {cases}
\delta_{k i} & : i \ne s \\
\delta_{k s} + \lambda \delta_{k t} & : i = s
\end {cases}$
where $\delta$ denotes the Kronecker delta.
Then:
=== Case $3$ ===
Let $e$ be the elementary column operation $\kappa_s \leftrightarrow \kappa_t$:
By Exchange of Columns as Sequence of Other Elementary Column Operations, this elementary column operation can be expressed as:
:$\paren {e_1 e_2 e_3 e_4 \mathbf A} = e \paren {\mathbf A}$
where the $e_i$ are elementary column operations of the other two types.
For each $e_i$, let $\mathbf E_i = e_i \paren {\mathbf I}$.
Then:
== Also see ==
* Elementary Row Operations as Matrix Multiplications
== Sources ==
* 
Category:Conventional Matrix Multiplication
Category:Elementary Column Operations
Category:Elementary Matrices
Elementary Column Operations as Matrix Multiplications/Corollary

== Theorem ==
Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary column operation.
Then there exists an elementary column matrix $\mathbf E$ of order $n$ such that:
:$\mathbf X \mathbf E = \mathbf Y$
== Proof ==
Let $e$ be the elementary column operation such that $e \paren {\mathbf X} = \mathbf Y$.
Then this result follows immediately from Elementary Column Operations as Matrix Multiplications:
:$e \paren {\mathbf X} = \mathbf X \mathbf E = \mathbf Y$
where $\mathbf E = e \paren {\mathbf I}$.
Category:Conventional Matrix Multiplication
Category:Elementary Column Operations
Elementary Function/Examples

== Examples of Elementary Functions ==
Examples of elementary functions include:
:$2 x - \ln x + \dfrac {e^{\sin x} } x$
:$\paren {\arccos x}^2 + \dfrac {e^{x^2 + 2x + 1} } {\sqrt {3 x} } - \map \ln {\ln 4 x}$
== Sources ==
* : Chapter $1$: Basic Concepts: Lesson $2 \text E$: The Elementary Functions
Category:Examples of Elementary Functions
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 ===
=== Case $(2)$: Add Scalar Product of Column to Another ===
=== Case $(3)$: Exchange Columns ===
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
Category:Elementary Column Operations
Category:Elementary Matrices
Category:Elementary Matrix corresponding to Elementary Column Operation
Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns

== 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:
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:
:$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.
By definition, $\mathbf E$ is the square matrix of order $n$ 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.
Category:Elementary Matrix corresponding to Elementary Column Operation
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:
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}$
== Proof ==
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.
Category:Elementary Matrix corresponding to Elementary Column Operation
Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add

== 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:
for $1 \le i \le m$, $1 \le j \le m$.
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:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{b i} \cdot \delta_{j a}$
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:
:$I_{a b}$ 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 $i$ of $\mathbf I$ are to have the corresponding elements of column $j$ added to them after the latter have been multiplied by $\lambda$.
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 $b \ne i$, $E_{a b} = \delta_{a b}$
:where $b = i$:
::$E_{a b} = \delta_{a b}$ where $a \ne j$
::$E_{a b} = \delta_{a b} + \lambda \cdot 1$ where $a = j$
That is:
:$E_{a b} = \delta_{a b}$ for all elements of $\mathbf E$
except where $b = i$ and $a = j$, at which element:
:$E_{a b} = \delta_{a b} + \lambda$
That is:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{b i} \cdot \delta_{j a}$
Hence the result.
== Also presented as ==
This can also be seen presented as:
:$E_{a b} = \begin{cases} \delta_{a b} & : b \ne i \\ \delta_{i a} + \lambda \delta_{j a} & : b = i \end{cases}$
when it is considered desirable to make the nature of the columns more easily understood.
Category:Elementary Matrix corresponding to Elementary Column Operation
Elementary Matrix corresponding to Elementary Row Operation

== Theorem ==
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.
Let $e$ be an elementary row operation on $\mathbf I$.
Let $\mathbf E$ be the elementary row matrix of order $m$ uniquely defined as:
:$\mathbf E = e \paren {\mathbf I}$
where $\mathbf I$ is the unit matrix.
Let $r_k$ denote the $k$th row of $\mathbf I$ for $1 \le k \le m$. 
=== Case $(1)$: Scalar Product of Row ===
=== Case $(2)$: Add Scalar Product of Row to Another ===
=== Case $(3)$: Exchange Rows ===
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 Column Operation
Category:Elementary Row Operations
Category:Elementary Matrices
Category:Elementary Matrix corresponding to Elementary Row Operation
Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows

== Theorem ==
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.
Let $e$ be the elementary row operation acting on $\mathbf I$ as:
for $1 \le i \le m$, $1 \le j \le m$.
Let $\mathbf E$ be the elementary row matrix defined as:
:$\mathbf E = e \paren {\mathbf I}$
$\mathbf E$ is the square matrix of order $m$ of the form:
:$E_{a b} = \begin {cases} \delta_{a b} & : \text {if $a \ne i$ and $a \ne j$} \\ \delta_{j b} & : \text {if $a = i$} \\ \delta_{i b} & : \text {if $a = j$} \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:
:$I_{a b}$ 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 row $i$ of $\mathbf I$ are to be exchanged with the corresponding elements of row $j$.
By definition of unit matrix:
:all elements of row $i$ are $0$ except for element $I_{i i}$, which is $1$.
:all elements of row $j$ 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 rows except $i$ and $j$ are unchanged)
:where $a = i$, $E_{a b} = \delta_{j b}$ (the contents of row $j$)
:where $a = j$, $E_{a b} = \delta_{i b}$ (the contents of row $i$)
That is:
:$E_{a b} = \begin {cases} \delta_{a b} & : \text {if $a \ne i$ and $a \ne j$} \\ \delta_{j b} & : \text {if $a = i$} \\ \delta_{i b} & : \text {if $a = j$} \end {cases}$
Hence the result.
Category:Elementary Matrix corresponding to Elementary Row Operation
Elementary Matrix corresponding to Elementary Row Operation/Scale Row

== Theorem ==
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.
Let $e$ be the elementary row operation acting on $\mathbf I$ as:
for $1 \le k \le m$.
Let $\mathbf E$ be the elementary row matrix defined as:
:$\mathbf E = e \paren {\mathbf I}$
$\mathbf E$ is the square matrix of order $m$ 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}$
== Proof ==
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 row $k$ of $\mathbf I$ are to be multiplied by $\lambda$.
By definition of unit matrix, all elements of row $k$ 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 rows of $\mathbf E$ are the same as the corresponding elements of $\mathbf I$.
Hence the result.
Category:Elementary Matrix corresponding to Elementary Row Operation
Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add

== Theorem ==
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$.
Let $e$ be the elementary row operation acting on $\mathbf I$ as:
for $1 \le i \le m$, $1 \le j \le m$.
Let $\mathbf E$ be the elementary row matrix defined as:
:$\mathbf E = e \paren {\mathbf I}$
$\mathbf E$ is the square matrix of order $m$ of the form:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
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:
:$I_{a b}$ 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 row $i$ of $\mathbf I$ are to have the corresponding elements of row $j$ added to them after the latter have been multiplied by $\lambda$.
By definition of unit matrix:
:all elements of row $i$ are $0$ except for element $I_{i i}$, which is $1$.
:all elements of row $j$ are $0$ except for element $I_{j j}$, which is $1$.
Thus in $\mathbf E$:
:where $a \ne i$, $E_{a b} = \delta_{a b}$
:where $a = i$:
::$E_{a b} = \delta_{a b}$ where $b \ne j$
::$E_{a b} = \delta_{a b} + \lambda \cdot 1$ where $b = j$
That is:
:$E_{a b} = \delta_{a b}$ for all elements of $\mathbf E$
except where $a = i$ and $b = j$, at which element:
:$E_{a b} = \delta_{a b} + \lambda$
That is:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
Hence the result.
== Also presented as ==
This can also be seen presented as:
:$E_{a b} = \begin{cases} \delta_{a b} & : a \ne i \\ \delta_{i b} + \lambda \delta_{j b} & : a = i \end{cases}$
when it is considered desirable to make the nature of the rows more easily understood.
Category:Elementary Matrix corresponding to Elementary Row Operation