Let $A$ be an $n$ by $n$ real matrix of order $d$. i.e. $d$ is the smallest positive integer greater than $1$ that makes $A^{d}=I_{n}$. The set of trace zero real matrices form $n^{2}-1$ dimensional real vector space. Let us call this vector space $V$. Let $\phi_{A}$ be the map from $V$ to $V$ sending $X$ to $AXA^{-1}$. Then $\phi$ is a linear transformation of $V$. I would like to know that if there is known relationship between the trace of $A$ and the trace of $\phi_{A}$.
Since $A$ is of finite order $A$ is conjugate to the direct sum of $2$ by $2$ rotation matrices $B_{\theta_{i}}$s, some identity matrix $I_{k_{1}}$ and some $-$identity matrix $-I_{k_{2}}$. Here the rotation matrix $B_{\theta_{i}}$ is the matrix \begin{displaymath} \left( \begin{array}{cc} \cos \theta_{i} & -\sin \theta_{i} \ \sin \theta_{i} & \cos \theta_{i} \
\end{array} \right) \end{displaymath} where $\theta_{i}$ is a rational multiple of $2\pi$.
Since $\phi_{A}$ is a linear transormation of finite order, there exists basis of $V$ such that the matrix for $\phi_{A}$ is a form of direct sum of rotation, Identity and $-$identity matrices. But is there any relation between trace of $A$ and trace of $\phi_{A}$ in terms of the angles $\theta_{i}s$?
