Characterization of $k$-spectrally monomorphic Hermitian matrices

Abstract: 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 ${1,\ldots,min{k, n-k}}$.