Definition:Elementary Matrix Operation

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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