What is the general form of a real orthogonal matrix of $n$th order that satisfies the condition $A^p=E$ where $p$ is a given positive integer, and $E$ is the identity matrix?
As I see, in the case of $p=2$ we have $A^2=E$ whence $A^{-1}=A$, then by the definition of orthogonality $A^T=A^{-1}$, whence $A^T=A$, that is $A$ is a symmetric matrix.
And what can one say about the general view of such matrices, for example for $p=5$ or $p=8$? Is it possible to display this general view?
On
If $p=0$, any orthogonal $A$ will do. If $p\ne0$, the general form of $A$ is given by $$ A=Q\left(R\left(\frac{2k_1\pi}{p}\right)\oplus R\left(\frac{2k_2\pi}{p}\right)\oplus\cdots\oplus R\left(\frac{2k_m\pi}{p}\right)\oplus I_r\oplus-I_s\right)Q^T $$ where $Q$ is any real orthogonal matrix, the $k_i$s are any integers, $R(\theta)$ denotes the $2\times2$ rotation matrix for an angle $\theta$, and $m,r,s$ are nonnegative integers such that $2m+r+s=n$. In addition, when $p$ is odd, $s$ must be zero, or else $(-I_s)^p$ will cause a reflection.
According to wikipedia , any Special Orthogonal matrix $R$ can be written as the exponential of a skew symmetric matrix $A$ , $R=e^A$. So if you can find an $A$ such that $I=e^A$ then $R = e^{{1\over p}A}$ is the $p$'th root of the identity matrix. Furthermore, $A$ can be written as $Q\Sigma Q^T$ with $Q$ orthogonal and $\Sigma$ skew symmetric and block diagonal with blocks of order 1 or 2. selecting each block to be $2\pi \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}$ or $\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$ will ensure that indeed $I=e^A$.