An involution matrix $A$ is defined by the condition $$ A^2 = I \tag{1} $$
The eigenvalues of an involution matrix $A$ are the roots of unity.
Generalizing, an $m$-involution matrix $A$ is defined by the condition $$ A^m = I \tag{2} $$
The eigenvalues of an $m$-involution matrix $A$ are the $m$th roots of unity.
In a similar manner, an anti-involution matrix $A$ can be defined as $$ A^2 = - I $$
(I hope that this is the standard definition)
An example of an anti-involution matrix is $$ A = \left[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right] $$
The eigenvalues of the above matrix $A$ are: $\pm j$.
The characteristic equation of this $2 \times 2$ matrix $A$ is $$ \lambda^2 + 1 = 0 $$
Is it possible to show that the eigenvalues of an anti-involution matrix are the square roots of $-1$ ?
In a similar way, if we define anti-m-involution matrix $A$ as $$ A^m = - I, $$ can we show that its eigenvalues are the $m$th roots of $-1$?
I don't ask about the eigenvectors of anti-involution matrices as I am not aware of any results for the eigenvectors of involution matrices.. Kindly help me on their eigenvalues. Thank you.