  A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal
submatrices of its adjacency matrix have the same characteristic polynomial.
Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show
that there are no other for $k\in \{3,\ldots,n-3\} $. Furthermore, we prove
that for $n\geq 5$, a non-transitive $n$-tournament is $(n-2)$-spectrally
monomorphic if and only if it is doubly regular. Finally, we give some results
on $(n-1)$-spectrally monomorphic regular tournaments.
