nLab monomial matrix

A monomial matrix with entries in a field $F$ is an $n \times n$ matrix with exactly one nonzero entry in each row and exactly one nonzero entry in each column. Under matrix multiplication, monomial matrices form a group; this group is isomorphic to a wreath product

$$S_n \ltimes (F^\ast)^n$$

with respect to the canonical action of the permutation group $S_n$ on the $n^{th}$ power $(F^\ast)^n$ of the group of nonzero scalars.

The special case where the non-vanishing entries are 1 is permutation matrices