Definition:Elementary Operation/Row
Definition
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.
The elementary row operations on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows.
For some $i, j \in \closedint 1 m: i \ne j$:
| \((\text {ERO} 1)\) | $:$ | \(\ds r_i \to \lambda r_i \) | For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$ | ||||||
| \((\text {ERO} 2)\) | $:$ | \(\ds r_i \to r_i + \lambda r_j \) | For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$ | ||||||
| \((\text {ERO} 3)\) | $:$ | \(\ds r_i \leftrightarrow r_j \) | Exchange rows $i$ and $j$ |
Also defined as
The order of presentation of the elementary matrix operations, either row or column, may vary according to the source.
Some sources use the Greek letter $\rho$ to enumerate the rows, and $\kappa$ to enumerate the columns, and jocularly remind us that the name rho of the letter $\rho$ is pronounced row.
Examples
Example: $r_2 \to \lambda r_2$
Consider the elementary row operation $e$ defined as:
- $e := r_2 \to \lambda r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
- $\begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$
Example: $r_3 \to r_3 + 2 r_2$
Consider the elementary row operation $e$ defined as:
- $e := r_3 \to r_3 + 2 r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
- $\begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end {pmatrix}$
Example: $r_1 \leftrightarrow r_2$
Consider the elementary row operation $e$ defined as:
- $e := r_1 \leftrightarrow r_2$
acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.
The elementary row matrix corresponding to $e$ is:
- $\begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$
Also see
- Results about elementary row operations can be found here.
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Definition $1.2$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): elementary matrix operation (abbrev. E-operation)
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elementary matrix operation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): elementary row operation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): row operation