An square matrix over a field is unimodular if its determinant is 1. The set of all 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 any invertible matrix and is unimodular, then 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 over an -dimensional vector space (over a field ) 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 is a commutative ring with 1, and is an matrix over (entries are elements of ) with . Then is said to be unimodular if it can be “completed” to a square unimodular matrix over . By completion of to we mean that of the rows in are exactly the rows of . Of course, the operation of completion from a matrix to a square matrix can be done via columns too.
Let is an matrix and is any row of . If is unimodular, then is unimodular viewed as a matrix. A unimodular matrix is called a unimodular row, or a unimodular vector. A unimodular column can be defined via a similar procedure. Let be a matrix over . Then the unimodularity of means that
To see this, let be a completion of with . Since is a multilinear operator over the rows (or columns) of , we see that
|Date of creation||2013-03-22 14:57:50|
|Last modified on||2013-03-22 14:57:50|
|Last modified by||CWoo (3771)|
|Defines||unimodular linear transformation|