  We consider the action on $\mathbb{F}_2^n$ by cyclic permutations
($\mathbb{Z}/n\mathbb{Z}$). Two elements $x, y\in \mathbb{F}_2^n$ are in the
same orbit if they are cyclic shifts of each other. Cryptographic properties of
rotation symmetric Boolean functions can be efficiently computed using the
square matrix $_n\mathcal{A}$, the construction of which uses orbit
representatives of the cyclic shifting action. In 2018, Ciungu and Iovanov
proved that $_n\mathcal{A}^2=2^n\cdot I$, the identity matrix of dimension
$g_n\times g_n$ where $g_n$ is the number of orbits. In this paper, we answer
the open question of the precise number of positive and negative eigenvalues of
$_n\mathcal{A}$.
