Reduced Echeleon Forms
Reduced echeleon forms are a specialized form of Echeleon Forms which is:
$$ \begin{bmatrix} 0 & 1 & 0 & * & 0 \\0 & 0 & 1 & * & 0 \\0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & 0 \end{bmatrix} $$
The main difference between reduced and regular echeleon forms is that $P$s are $1$ and the column that the pivots are in, all the other rows must have the value $0$.