# monomial matrix

Let $A$ be a matrix with entries in a field $K$. If in every http://planetmath.org/node/2464row and every http://planetmath.org/node/2464column 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 and 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 http://planetmath.org/node/1176orthogonal. The only exception is when $K=\mathbb{F}_{2}$ (the finite field with $2$ elements), where the $n\times n$ monomial matrices and the $n\times n$ permutation matrices coincide.

Title monomial matrix MonomialMatrix 2013-03-22 15:15:51 2013-03-22 15:15:51 GrafZahl (9234) GrafZahl (9234) 5 GrafZahl (9234) Definition msc 20H20 msc 15A30 PermutationMatrix