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Following Wikipedia on Gaussian elimination, each row is represented by the letter $L$ with subscript row #, starting with 1 at the top, and increasing by 1 down each row.

Elimination notation

Let's say we have a Matrix:

$A={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}$

Multiplying the second row by $-4$ times the first row, resulting in

$A={\begin{bmatrix}1&2&3\\0&-3&-6\\7&8&9\end{bmatrix}}$

the elimination notation can be represented like this:

$E_{21}={\begin{bmatrix}1&0&0\\4&1&0\\0&0&1\end{bmatrix}}$ The row number where the operation is being applied comes first.

The first column represents usage of the first row in a row operation, the second column represents usage of the second row, et. cetera. Each operation makes up our original matrix $A$, which is why, with the first and third rows, which haven't been altered, the only operations are $1\cdot L_{1}$.

References

• https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/resources/lu-decomposition/