Definition:Elementary Operation/Column

Definition

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

• Definition:Column Equivalence
• Definition:Column Operation
• Definition:Elementary Column Matrix

• Definition:Elementary Row Operation
• Definition:Elementary Matrix Operation

• Results about elementary column 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)
• 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