Integer unimodular matrix

An integer unimodular matrix , in the corresponding context also only unimodular matrix , is in algebra a square matrix whose entries are all integers and whose determinant is or . This condition is equivalent to the fact that the entries are integer, the matrix can be inverted , and the inverse matrix also only has integer entries. The integer unimodular matrices with rows and columns form the general linear group with the matrix multiplication . ${\displaystyle 1}$ ${\displaystyle -1}$ ${\displaystyle n}$ ${\displaystyle GL(n,\mathbb {Z} )}$

definition

A square matrix is called unimodular if for its determinant ${\displaystyle A\in \mathbb {Z} ^{n\times n}}$

${\displaystyle \det A\in \{1,-1\}}$

applies.

Examples

The matrix

${\displaystyle A:={\begin{pmatrix}2&1\\5&3\end{pmatrix}}\in \mathbb {Z} ^{2\times 2}}$

is unimodular: your determinant is . The inverse matrix ${\displaystyle \det(A)=2\cdot 3-5\cdot 1=1}$

${\displaystyle A^{-1}={\begin{pmatrix}3&-1\\-5&2\end{pmatrix}}}$

is again integer and unimodular. Important classes of integer unimodular matrices permutation , for one entry per row and column , and monomial matrices in which one entry per row and column , or is, and all other entries are. ${\displaystyle 1}$ ${\displaystyle +1}$ ${\displaystyle -1}$ ${\displaystyle 0}$

properties

Every integer unimodular matrix is regular and its inverse is in turn integer and unimodular. The product of two unimodular matrices also results in a unimodular matrix due to the determinant product theorem ${\displaystyle A,B\in \mathbb {Z} ^{n\times n}}$

${\displaystyle \det(A\cdot B)=\det A\cdot \det B}$.

The integral unimodular matrices with a fixed number of rows and columns therefore form a group with the multiplication , the general linear group . In other words, it is the automorphism of the free Abelian group of rank , with component-wise addition. Also the Kronecker product of two unimodular matrices and results again in a unimodular matrix, because it applies ${\displaystyle GL(n,\mathbb {Z} )}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle A\in \mathbb {Z} ^{n\times n}}$ ${\displaystyle B\in \mathbb {Z} ^{m\times m}}$

${\displaystyle \det(A\otimes B)=(\det A)^{n}\cdot (\det B)^{m}}$.

Due to the property that the inverse of an integer, unimodular matrix is again integer, it is particularly true for all systems of equations with a vector that contains only integer values ​​that their solution is integer. ${\displaystyle A}$ ${\displaystyle A\cdot x=b}$ ${\displaystyle b}$

use

In solid-state physics and in particular in crystallography , integer unimodular matrices occur as transformations between primitive unit cells: An operation of the unimodular matrices on the can be selected in such a way that they map a given grid onto itself and each primitive unit cell of a grid onto one . Two primitive unit cells can be converted into one another via a unimodular matrix. ${\displaystyle \mathbb {R} ^{n}}$

generalization

In commutative algebra , matrices over commutative rings are considered , among other things . A matrix with integer entries is just a matrix over the ring of integers. It is generally true that a square matrix over a commutative ring with one is invertible if and only if its determinant is a unit , that is, if its determinant is invertible in the underlying ring. In the ring of integers, and are the only two units (that is, they are the only integers with an integer reciprocal ). The constructive proof is possible by using the adjuncts . ${\displaystyle 1}$ ${\displaystyle -1}$

See also

• Unimodular matrix (disambiguation)
• Totally unimodular matrix

Individual evidence

1. ↑ Shoon Kyung Kim: Group Theoretical Methods and Applications to Molecules and Crystals . Cambridge University Press , 2004, ISBN 0-511-03620-5 , pp. 297 ( online ).