Unimodular matrix

An integer unimodular matrix, in the appropriate context even unimodular matrix is a square matrix algebra whose entries are all integers and whose determinant is or. This condition is equivalent to the fact that the entries are integers, the matrix is ​​invertible, and the inverse matrix also has only integer entries. The integer unimodular matrices having rows and columns forming the linear array with the general matrix multiplication.

Definition

A square matrix is called unimodular if its determinant for

Applies.

Examples

The matrix

Is unimodular: Your determinant. The inverse matrix

Is again an integer. An important class of integer unimodular matrices are the permutation matrices.

Properties

Each integer unimodular matrix is regular and its inverse is again an integer unimodular. Also, the product of two unimodular matrices yields again a unimodular matrix, because it is the product set of determinants

Therefore, the integer unimodular matrices with a fixed number of rows and columns form a group with the multiplication, the overall linear array. In other words, it is the automorphism group of the free abelian group of rank, with componentwise addition. Also the Kronecker product of two matrices, and again results in a unimodular unimodular matrix, because it is

Use

In solid-state physics and in particular crystallography occur integer unimodular matrices as transformations between primitive unit cells: it can be chosen such an operation of unimodular matrices on the that they represent a given lattice to itself and each primitive unit cell of a grid, in turn, to such. The two primitive unit cells can be converted via a unimodular matrix together.

Generalization

In commutative algebra, among other matrices are considered over commutative rings. 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 unity is invertible if its determinant is a unit, that is, if its determinant is invertible in the underlying ring. Are in the ring of integers and the only two units ( that is, they are the only integers with an integer reciprocal). The proof is constructive possible by the use of adjuncts.