I am looking for a name of a following matrix property: each next row has more starting zeros than the last one. The last few rows can all be full of zeros.
I was trying to search for that but found nothing...
Regarding upper triangular/trapezoidal matrices - that is not what I want. For example $\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}$ is upper triangular but doesn't satisfy my property.
Regarding the reasons for why I'm looking for this: I think that LUP-decomposition can be generalized onto rectangular matrices with both factors not only being triangular, but also U having this property I described and L having a similar one for columns. Haven't found much on that side either...
I think you are looking for the term row echelon form. By Gauss elimination every matrix can be transformed into this form.