Let $A$ be a matrix with entries in a field $K$ If in every row and every column of $A$ there is exactly one nonzero entry, then $A$ is a monomial matrix.

Obviously, a monomial matrix is a square matrix and there exists a rearrangement of rows and columns such that the result is a diagonal matrix.

The $n\times n$ monomial matrices form a group under matrix multiplication. This group contains the $n\times n$ permutation matrices as a subgroup. A monomial matrix is invertible but, unlike a permutation matrix, not necessarily orthogonal. The only exception is when $K=\mbb{F}_2$ (the finite field with $2$ elements), where the $n\times n$ monomial matrices and the $n\times n$ permutation matrices coincide.