A matrix of a single 1 in each row and 0 elsewhere

Question

Is there a particular name given to a matrix of m rows and n columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance:

0 1 0 0 0 0 0
0 0 0 0 1 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 1
0 0 1 0 0 0 0

Note: It may or may not be a square matrix.

Answer 1

I'm not aware of a common name for this type of matrix. However, a nice way to describe it is that the rows of your matrix are rows of the identity matrix with possible repetition.

Answer 2

I haven't heard of a name for the class of such matrices. If I had to suggest a name for such a matrix, I would call it a "row selection matrix", since if $A \in R^{m\times n}$ is such a matrix and $B \in R^{n\times k}$ an arbitrary matrix, then every row of $AB$ is also a row of $B$.