# unimodular matrix

An $n\times n$ square matrix^{} over a field is *unimodular* if its determinant^{} is 1. The set of all $n\times n$ unimodular matrices^{} forms a group under the usual matrix multiplication^{}. This group is known as the special linear group^{}. Any of its subgroup^{} is simply called a *unimodular group ^{}*. Furthermore, unimodularity is preserved under similarity transformations

^{}: if $S$ any $n\times n$ invertible matrix and $U$ is unimodular, then ${S}^{-1}US$ is unimodular. In view of the last statement, the special linear group is a normal subgroup

^{}of the group of all invertible matrices, known as the general linear group

^{}.

A linear transformation $T$ over an $n$-dimensional vector space^{} $V$ (over a field $F$) is *unimodular* if it can be represented by a unimodular matrix.

The concept of the unimodularity of a square matrix over a field can be readily extended to that of a square matrix over a commutative ring. Unimodularity in square matrices can even be extended to arbitrary finite-dimensional matrices: suppose $R$ is a commutative ring with 1, and $M$ is an $m\times n$ matrix over $R$ (entries are elements of $R$) with $m\le n$. Then $M$ is said to be *unimodular* if it can be “completed” to a $n\times n$ square unimodular matrix $N$ over $R$. By completion of $M$ to $N$ we mean that $m$ of the $n$ rows in $N$ are exactly the rows of $M$. Of course, the operation of completion from a matrix to a square matrix can be done via columns too.

Let $M$ is an $m\times n$ matrix and $v$ is any row of $M$. If $M$ is unimodular, then $v$ is unimodular viewed as a $1\times n$ matrix. A $1\times n$ unimodular matrix is called a *unimodular row*, or a *unimodular vector*. A $n\times 1$ *unimodular column* can be defined via a similar^{} procedure. Let $v=({v}_{1},\mathrm{\dots},{v}_{n})$ be a $1\times n$ matrix over $R$. Then the unimodularity of $v$ means that

$${v}_{1}R+\mathrm{\cdots}+{v}_{n}R=R.$$ |

To see this, let $U$ be a completion of $v$ with $\mathrm{det}(U)=1$. Since $\mathrm{det}$ is a multilinear operator over the rows (or columns) of $U$, we see that

$$1=\mathrm{det}(U)={v}_{1}{r}_{1}+\mathrm{\cdots}+{v}_{n}{r}_{n}.$$ |

Title | unimodular matrix |

Canonical name | UnimodularMatrix |

Date of creation | 2013-03-22 14:57:50 |

Last modified on | 2013-03-22 14:57:50 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 13 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 20H05 |

Classification | msc 15A04 |

Classification | msc 15A09 |

Related topic | SpecialLinearGroup |

Defines | unimodular linear transformation |

Defines | unimodular row |

Defines | unimodular column |

Defines | unimodular group |

Defines | unimodular vector |