Category:Definitions/Elementary Matrix Operations

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This category contains definitions related to Elementary Matrix Operations.
Related results can be found in Category:Elementary Matrix Operations.


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 $j$             

Subcategories

This category has the following 3 subcategories, out of 3 total.