Definition:Elementary Matrix Operation
Definition
Elementary Row Operation
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 $j$ |
Elementary Column Operation
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.
The elementary column operations on $\mathbf A$ are operations which act upon the columns of $\mathbf A$ as follows.
For some $i, j \in \closedint 1 n: i \ne j$:
\((\text {ECO} 1)\) | $:$ | \(\ds \kappa_i \to \lambda \kappa_i \) | For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$ | ||||||
\((\text {ECO} 2)\) | $:$ | \(\ds \kappa_i \to \kappa_i + \lambda \kappa_j \) | For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$ | ||||||
\((\text {ECO} 3)\) | $:$ | \(\ds \kappa_i \leftrightarrow \kappa_j \) | Interchange columns $i$ 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.
Also see
- Results about elementary matrix operations can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): elementary matrix operation (abbrev. E-operation)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elementary matrix operation