Linear and multilinear algebra; matrix theory Definition

Elementary matrix

Builds on Identity matrix Elementary row operations

Definition

Let $F$ be a field and let $n \ge 1$ be an integer. An $n \times n$ elementary matrix over $F$ is a matrix that results from the identity matrix $I_n$ by one elementary row operation. There are three types:

$P_{ij}$ , for $i \ne j$ : exchange rows $i$ and $j$ of $I_n$ .
$D_i(c)$ , for $c \ne 0$ : multiply row $i$ of $I_n$ by $c$ .
$T_{ij}(c)$ , for $i \ne j$ and $c \in F$ : add $c$ times row $j$ of $I_n$ to row $i$ .

Remarks

For each $n \times p$ matrix $A$ , the product $EA$ is the result of the same row operation on $A$ . The product $AE$ performs the matching column operation on $A$ . Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type. Each invertible matrix over a field is a product of elementary matrices.

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Identity matrix object
Matrix definition
Elementary row operations definition
Elementary matrix definition

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Tag	`0M79HRJCRR`
URI	`m2://0M79HRJCRR`
Kind	definition
Writing	W0 stub
MSC	15A23
Defines	Elementary matrix
Uses	Identity matrix, Elementary row operations

Elementary matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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