# 9: Transformation matrices

- Page ID
- 9333

Matrices can be used to map one set of coordinates or functions onto another set. Matrices used for this purpose are called *transformation matrices*. In group theory, we can use transformation matrices to carry out the various symmetry operations considered at the beginning of the course. As a simple example, we will investigate the matrices we would use to carry out some of these symmetry operations on a vector \(\begin{pmatrix} x, y \end{pmatrix}\).

## The identity Operation

The identity operation leaves the vector unchanged, and as you may already suspect, the appropriate matrix is the identity matrix.

\[\begin{pmatrix} x, y \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} x, y \end{pmatrix} \label{9.1}\]

## Reflection in a plane

The simplest example of a reflection matrix corresponds to reflecting the vector \(\begin{pmatrix} x, y \end{pmatrix}\) in either the \(x\) or \(y\) axes. Reflection in the \(x\) axis maps \(y\) to \(-y\), while reflection in the \(y\) axis maps \(x\) to \(-x\). The appropriate matrix is very like the identity matrix but with a change in sign for the appropriate element. Reflection in the \(x\) axis transforms the vector \(\begin{pmatrix} x, y \end{pmatrix}\) to \(\begin{pmatrix} x, -y \end{pmatrix}\), and the appropriate matrix is

\[\begin{pmatrix} x, y \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} x, -y \end{pmatrix} \label{9.2}\]

**Figure \(\PageIndex{1}\)**: Reflection across the x-axis

Reflection in the y axis transforms the vector \(\begin{pmatrix} x, y \end{pmatrix}\) to \(\begin{pmatrix} -x, y \end{pmatrix}\), and the appropriate matrix is

\[\begin{pmatrix} x, y \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -x, y \end{pmatrix} \label{9.3}\]

**Figure \(\PageIndex{2}\)**: Reflection across the y-axis

More generally, matrices can be used to represent reflections in any plane (or line in 2D). For example, reflection in the 45° axis shown below maps

\(\begin{pmatrix} x, y \end{pmatrix}\) onto \(\begin{pmatrix} -y, -x \end{pmatrix}\).

\[\begin{pmatrix} x, y \end{pmatrix} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} -y, -x \end{pmatrix} \label{9.4}\]

**Figure \(\PageIndex{3}\)**: Reflection across the axis that is rotated 45° with with respect to x-axis.

## Rotation about an Axis

In two dimensions, the appropriate matrix to represent rotation by an angle \(\theta\) about the origin is

\[R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \label{9.5}\]

In three dimensions, rotations about the \(x\), \(y\) and \(z\) axes acting on a vector \(\begin{pmatrix} x, y, z \end{pmatrix}\) are represented by the following matrices.

\[R_{x}(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix} \label{9.6a}\]

\[R_{y}(\theta) = \begin{pmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{pmatrix} \label{9.6b}\]

\[R_{z}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \label{9.6c}\]

## Contributors

Claire Vallance (University of Oxford)