Question

Write the four rotation matrices for rotations of vectors in the xy plane through angles $\mathbf{90}\mathbf{°}\mathbf{,}\mathbf{180}\mathbf{°}\mathbf{,}\mathbf{270}\mathbf{°}\mathbf{,}\mathbf{360}\mathbf{°}\mathbf{}\mathbf{(}\mathbf{or}\mathbf{}\mathbf{0}\mathbf{°}\mathbf{)}$[see equation (7.12)]. Verify that these 4 matrices under matrix multiplication satisfy the four group requirements and are a matrix representation of the cyclic group of order 4. Write their multiplication table and compare with Equations (13.1) and (13.2).

Short answer

The matrices of rotation in two dimensions form a group and are a representation of the cyclic group of order 4.

Step-by-step solution

Given information

The xy plane through angles$90°,180°,270°,360°(\mathrm{or}0°)$

Rotation matrix

A rotation matrix can be defined as a transformation matrix that operates on a vector and produces a rotated vector such that the coordinate axes always remain fixed.

Representation of the cyclic group.

In this problem the matrices of rotation in the xy plane. A general rotation matrix by an angle$\theta$is written as

$$\begin{gathered} R\left(\theta \right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \\ \mathrm{For}\mathrm{\theta }=90°,180°,270°,360°, \\ R\left(360°\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\mathrm{I}, \\ \mathrm{R}\left(90°\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right), \\ \mathrm{R}\left(180°\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)=-\mathrm{I}, \\ \mathrm{R}\left(270°\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right), \end{gathered}$$

The rotation by $0°$is the unit element. By multiplying the element with each other (multiplication by $R\left(360°\right)$) is trivial), obtain

$$\begin{gathered} \mathrm{R}\left(90°\right)\mathrm{R}\left(270°\right)=\mathrm{I}=\mathrm{R}\left(270°\right)\mathrm{R}\left(90°\right) \\ \mathrm{R}\left(90°\right)\mathrm{R}\left(180°\right)=\mathrm{R}\left(270°\right)\mathrm{R}\left(180°\right)\mathrm{R}\left(90°\right) \\ \mathrm{R}\left(180°\right)\mathrm{R}\left(270°\right)=\mathrm{R}\left(90°\right)=\mathrm{R}\left(270°\right)\mathrm{R}\left(180°\right) \\ \mathrm{R}\left(90°\right)\mathrm{R}\left(270°\right)=\mathrm{R}\left(180°\right) \\ \mathrm{Solve}\mathrm{further} \\ \mathrm{R}\left(180°\right)\mathrm{R}\left(180°\right)=\mathrm{R}\left(0\right), \\ \mathrm{R}\left(270°\right)\mathrm{R}\left(270°\right)=\mathrm{R}\left(180°\right). \end{gathered}$$

From this the property of closure is satisfied, as well as the property of the existence of the inverse, that is, the inverse of $\mathrm{R}\left(90°\right)$is$\mathrm{R}\left(270°\right)$ and vice versa,$\mathrm{R}\left(180°\right)$is its own inverse, as well as the identity. Also verify associativity

$$\begin{gathered} \left(\mathrm{R}\left(90°\right)\mathrm{R}\left(180°\right)\right)\mathrm{R}\left(270°\right) \\ =\mathrm{R}\left(270°\right)\mathrm{R}\left(270°\right) \\ =\mathrm{R}\left(180°\right) \\ =\mathrm{R}\left(270°\right)\mathrm{R}\left(270°\right)\mathrm{R}\left(270°\right)) \\ =\mathrm{R}\left(90°\right)\mathrm{R}\left(90°\right) \\ =\mathrm{R}\left(180°\right) \end{gathered}$$

The past two equations give the same result. This is true because multiplication of two rotation matrices gives a rotation matrix by an angle which is the sum of the angles in the product. The number addition is associative.

Also, this means write these matrices as

role="math" localid="1659341122161" $$\begin{gathered} \mathrm{R}\left(90°\right)\mathrm{R}\left(180°\right)\mathrm{R}{\left(270°\right)}^2,\mathrm{R}\left(90°\right) \\ =\mathrm{R}{\left(90°\right)}^3,\mathrm{R}\left(360°\right) \\ =\mathrm{R}{\left(90°\right)}^4 \\ =\mathrm{I} \end{gathered}$$

This is a representation of the cyclic group.

Write the multiplication table

Write the multiplication table

Obtain the same table as for the representations $\pm 1,\pm i$and $0,\pi /2,\pi ,3\pi /2,$, identify .

$R\left(360°\right)~1~0,R\left(900°\right)~i~\pi /2,R\left(180°\right)~-1~\pi ,R\left(270°\right)~-i~3\pi /2.$