# 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 |
---|---|

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 |