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unimodular matrix (Definition)

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

$\displaystyle v_1R+\cdots+v_nR=R.$
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
$\displaystyle 1=\operatorname{det}(U)=v_1r_1+\cdots+v_nr_n.$



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See Also: special linear group

Also defines:  unimodular linear transformation, unimodular row, unimodular column, unimodular group, unimodular vector
Keywords:  unimodular, unimodularity
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Cross-references: operator, multilinear, columns, operation, rows, completion, square, finite-dimensional, commutative ring, vector space, linear transformation, general linear group, normal subgroup, matrix, invertible, similarity transformations, subgroup, special linear group, matrix multiplication, group, determinant, field, square matrix
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This is version 10 of unimodular matrix, born on 2005-01-26, modified 2006-09-09.
Object id is 6662, canonical name is UnimodularMatrix.
Accessed 6761 times total.

Classification:
AMS MSC15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
 20H05 (Group theory and generalizations :: Other groups of matrices :: Unimodular groups, congruence subgroups)

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