monomial matrix

Let $A$ be a matrix with entries in a field $K$ . If in every http://planetmath.org/node/2464row and everyhttp://planetmath.org/node/2464column of $A$ there is exactly one nonzero entry, then $A$ is amonomial matrix.

Obviously, a monomial matrix is a square matrix and there exists arearrangement of and such that the result is a diagonalmatrix.

The $n\\times n$ monomial matrices form a group under matrixmultiplication. This group contains the $n\\times n$ permutationmatrices as a subgroup. A monomial matrix is invertible but, unlike apermutation matrix, not necessarily http://planetmath.org/node/1176orthogonal. The only exception iswhen $K=\\mathbb{F}_{2}$ (the finite field with $2$ elements), where the $n\\times n$ monomial matrices and the $n\\times n$ permutation matricescoincide.

单词	MonomialMatrix
释义	monomial matrix Let $A$ be a matrix with entries in a field $K$ . If in every http://planetmath.org/node/2464row and everyhttp://planetmath.org/node/2464column of $A$ there is exactly one nonzero entry, then $A$ is amonomial matrix. Obviously, a monomial matrix is a square matrix and there exists arearrangement of and such that the result is a diagonalmatrix. The $n\\times n$ monomial matrices form a group under matrixmultiplication. This group contains the $n\\times n$ permutationmatrices as a subgroup. A monomial matrix is invertible but, unlike apermutation matrix, not necessarily http://planetmath.org/node/1176orthogonal. The only exception iswhen $K=\\mathbb{F}_{2}$ (the finite field with $2$ elements), where the $n\\times n$ monomial matrices and the $n\\times n$ permutation matricescoincide.
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