monomial matrix

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

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

The n×n monomial matrices form a group under matrix multiplicationMathworldPlanetmath. This group contains the n×n permutation matricesMathworldPlanetmath as a subgroupMathworldPlanetmathPlanetmath. A monomial matrix is invertiblePlanetmathPlanetmathPlanetmath but, unlike a permutation matrix, not necessarily The only exception is when K=𝔽2 (the finite field with 2 elements), where the n×n monomial matrices and the n×n permutation matrices coincide.

Title monomial matrix
Canonical name MonomialMatrix
Date of creation 2013-03-22 15:15:51
Last modified on 2013-03-22 15:15:51
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 5
Author GrafZahl (9234)
Entry type Definition
Classification msc 20H20
Classification msc 15A30
Related topic PermutationMatrix