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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 matrixMathworldPlanetmath and there exists arearrangement of and such that the result is a diagonalmatrixMathworldPlanetmath.

The n×n monomial matrices form a group under matrixmultiplicationMathworldPlanetmath. This group contains the n×n permutationmatricesMathworldPlanetmath as a subgroupMathworldPlanetmathPlanetmath. A monomial matrix is invertiblePlanetmathPlanetmathPlanetmath but, unlike apermutation matrix, not necessarily http://planetmath.org/node/1176orthogonalMathworldPlanetmathPlanetmath. The only exception iswhen K=𝔽2 (the finite field with 2 elements), where then×n monomial matrices and the n×n permutation matricescoincide.

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更新时间:2025/5/4 7:20:40