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 $m\leq n$. 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 $v=(v_{1},\ldots,v_{n})$ be a $1\times n$ matrix over $R$. Then the unimodularity of $v$ means that

 $v_{1}R+\cdots+v_{n}R=R.$

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

 $1=\operatorname{det}(U)=v_{1}r_{1}+\cdots+v_{n}r_{n}.$
 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