unimodular matrix

An n×n square matrixMathworldPlanetmath over a field is unimodular if its determinantMathworldPlanetmath is 1. The set of all n×n unimodular matricesMathworldPlanetmath forms a group under the usual matrix multiplicationMathworldPlanetmath. This group is known as the special linear groupMathworldPlanetmath. Any of its subgroupMathworldPlanetmathPlanetmath is simply called a unimodular groupMathworldPlanetmath. Furthermore, unimodularity is preserved under similarity transformationsMathworldPlanetmath: if S any n×n invertible matrix and U is unimodular, then S-1US is unimodular. In view of the last statement, the special linear group is a normal subgroupMathworldPlanetmath of the group of all invertible matrices, known as the general linear groupMathworldPlanetmath.

A linear transformation T over an n-dimensional vector spaceMathworldPlanetmath 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×n matrix over R (entries are elements of R) with mn. Then M is said to be unimodular if it can be “completed” to a n×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×n matrix and v is any row of M. If M is unimodular, then v is unimodular viewed as a 1×n matrix. A 1×n unimodular matrix is called a unimodular row, or a unimodular vector. A n×1 unimodular column can be defined via a similarMathworldPlanetmath procedure. Let v=(v1,,vn) be a 1×n matrix over R. Then the unimodularity of v means that


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

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