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单词 UnimodularMatrix
释义

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

v1R++vnR=R.

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

1=det(U)=v1r1++vnrn.
Titleunimodular matrix
Canonical nameUnimodularMatrix
Date of creation2013-03-22 14:57:50
Last modified on2013-03-22 14:57:50
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id13
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 20H05
Classificationmsc 15A04
Classificationmsc 15A09
Related topicSpecialLinearGroup
Definesunimodular linear transformation
Definesunimodular row
Definesunimodular column
Definesunimodular group
Definesunimodular vector
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更新时间:2025/5/4 22:39:56