Is there a name for the following type of block matrices?
A matrix $A$ is [insert name here] if it can be decomposed into non-zero non-scalar submatrices such that each sub-matrix $B$, with $B$ spanning rows $i,\ldots,j$ and columns $k,\ldots,\ell$ (in $A$), is such that $A_{r,s} = 0$ for any $r,s$ such that $r < i$ and $s \in \{k, \ldots,\ell\}$, $r > j$ and $s \in \{k, \ldots,\ell\}$, $s < k$ and $r \in \{ i,\ldots,j\}$ or $s > \ell$ and $r \in \{ i, \ldots, j \}$.
In all places other than these non-zero submatrices, the matrix is 0, as implied. The submatrices need to cover all the non-zero values.
The idea is that the matrix can be decomposed into blocks such that anywhere above the block, to the left, or right, the matrix is 0.
If there is no specific name for this type of matrices, is there anything else that is known about them?