  This study explores the relationship between hypergraph automorphisms and the
spectral properties of matrices associated with hypergraphs. For an
automorphism $f$, an \( f \)-compatible matrices capture aspects of the
symmetry, represented by \( f \), within the hypergraph. First, we explore
rotation, a specific kind of automorphism and find that the spectrum of any
matrix compatible with a rotation can be decomposed into the spectra of smaller
matrices associated with that rotation. We show that the spectrum of any
\(f\)-compatible matrix can be decomposed into the spectra of smaller matrices
associated with the component rotations comprising \( f \). Further, we study a
hypergraph symmetry termed unit-automorphism, which induces bijections on the
hyperedges, though not necessarily on the vertex set. We show that unit
automorphisms also lead to the spectral decomposition of compatible matrices.
