PlanetMath: monomial matrix

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monomial matrix

(Definition)

Let be a matrix with entries in a field . If in every row and every column of there is exactly one nonzero entry, then 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 $K=\mathbb{F}_2$ (the finite field with elements), where the $n\times n$ monomial matrices and the $n\times n$ permutation matrices coincide.

"monomial matrix" is owned by GrafZahl.

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See Also: permutation matrix

Keywords:

matrix, monomial, group

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Cross-references: finite field, invertible, subgroup, permutation matrices, contains, matrix multiplication, group, diagonal matrix, square matrix, field, matrix
There are 2 references to this entry.

This is version 2 of monomial matrix, born on 2005-05-13, modified 2005-05-16.
Object id is 7050, canonical name is MonomialMatrix.
Accessed 3886 times total.

Classification:

AMS MSC:	15A30 (Linear and multilinear algebra; matrix theory :: Algebraic systems of matrices)
	20H20 (Group theory and generalizations :: Other groups of matrices :: Other matrix groups over fields)

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