  This paper solves the following problem about Hermitian matrices related to
the theory of $2$-structures:\emph{ }Let $n$ be a positive integer and $k$ be
an integer with $k\in \{3,\ldots,n-3\}$. Characterize the Hermitian matrices
$A$ such that the characteristic polynomials of the $k\times k$ submatrices of
$A$ are all equal. Such matrices are called $k$-spectrally monomorphic. A
crucial step to obtain this characterization is proving that if a matrix $A$ is
$k$-spectrally monomorphic then it is $l$-spectrally monomorphic for $l$ in
$\{1,\ldots,min\{k, n-k\}\}$.
