Properties of the rotation matrix

Question

$$R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$

Consider the above rotation matrix.

$a)$ Show that $R(\theta)$ is non-singular with: $R^{-1}(\theta) = R(-\theta)$

$b)$ Show that $R(\theta)^T = R(-\theta)$

$c)$ For what angles $\theta \in \mathbb{R}$ is $R(\theta)$ symmetric?

I only somewhat understand part $b$. I know that a skew-matrix is one in which its transpose is also its negative. Apart from that I'm not sure how to implement that into the question and I have no idea how to do part $a$ and $c$. I was wondering if someone could help me get started.

Answer 1

Hint:

(a) $\quad$ Calculate $R(-\theta)R(\theta)$

(c) $\quad$ By definition: symmetric when $R(\theta)^T=R(\theta)$

$\quad \quad \,$ By $(b)$ then $R(\theta)=R(-\theta) \implies \sin \theta=-\sin \theta$

Answer 2

a

A quick singularity check: $$ \det \mathbf{R} \left( \theta \right) = \cos^{2} \left( \theta \right) + \sin^{2} \left( \theta \right) = 1 $$ Because the determinant is not $0$, the rotation matrix is not singular. Is $$ \mathbf{R} \left( \theta \right)^{-1} = \mathbf{R} \left( -\theta \right)? $$ Check via multiplication $$ \begin{align} \mathbf{R} \left( \theta \right) \mathbf{R} \left( -\theta \right) &= % \left[ \begin{array}{rr} \cos ( \theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{array} \right] % \left[ \begin{array}{rr} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \\ \end{array} \right] \\[5pt] % &= % \left[ \begin{array}{cc} \cos ^2(\theta )+\sin ^2(\theta ) & 0 \\ 0 & \cos ^2(\theta )+\sin ^2(\theta ) \\ \end{array} \right] \\[5pt] % &= % \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right] \\[2pt] % &= \mathbf{I}_{2} % \end{align} $$

b

Show that the transpose matrix is $\mathbf{R}\left( -\theta \right)$. $$ \begin{align} \mathbf{R}\left( -\theta \right) &= \left[ \begin{array}{rr} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \\ \end{array} \right] \\[3pt] % \mathbf{R}^{T}\left( \theta \right) &= \left[ \begin{array}{rr} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \\ \end{array} \right] \\ \end{align} % $$

c

When does $\mathbf{R} \left( \theta \right)$ have a symmetric form like $$ \left[ \begin{array}{rr} a & b \\ b & b \\ \end{array} \right] ? $$ The diagonal entries are the same, $\cos \left( \theta \right)$. They pose no constraint. For the off-diagonal entries, the question is when does $$ \sin \left( \theta \right) = - \sin \left( \theta \right)? $$ The answer is when $\sin \left( \theta \right) = 0?$ which occurs for $$ \theta = 2k\pi, \quad k\in\mathbb{Z}. $$