Matrix representation of symmetry operations

Using carthesian coordinates (x,y,z) or some **position vector**, we are able to define an initial position of a point or an atom.

The initial vector is submitted to a symmetry operation and thereby transformed into some resulting vector defined by the coordinates x', y' and z'. In an algebraic context, this transformation is expressed a matrix which processes the initial position vector. We write

*final vector = Matrix * initial vector.*

The most primitive symmetry operation is the identity and yields a final vector identical to the initial vector. It is the **unity matrix** or **identity matrix** which leaves all coordiates unaffected.

If we want to perform a **reflection** on the xy-plane (analogous to a horizontal plane σ_{h}), coordinate z changes the sign.

The matrices which are applied for performing a reflection on the yz-plane
and xz-plane are the matrices σ_{x} and σ_{y} respectively.

The **inversion i** relates the coordinates (x,y,z) with (-x,-y,-z) and is connected with the following matrix:

Obviously, a twofold application of the inversion matrix yields the coordinates of the initial point (x,y,z) which is reflected by ** E = i*i**.

The matrix for a rotation about axis z by an arbitrary angle Θ is derived easily if we imagine two two-dimensional coordinate planes with identical origin but an angular difference of Θ between the axes. In our context of symmetry, we just need to deal with the discrete values of Θ = 2π/n for the angle of rotation.

The matrices for the symmetry operations *C _{2}(z),
C_{3}(z), C_{4}(z), C_{5}(z)* and

As we know rotatory-reflection to be a combination of rotation and reflection, a matrix representation for this operation is easily to be derived. For instance, to obtain the matrix for rotatory reflection S_{n}(z) we multiply the matrices for the fundamental operations &sigma_{z} and C_{n}.