Characterization of $k$-spectrally monomorphic Hermitian matrices

TL;DR

This paper characterizes Hermitian matrices where all $k$-by-$k$ submatrices share the same characteristic polynomial, revealing their spectral uniformity and extending understanding of $k$-spectrally monomorphic matrices.

Contribution

It provides a complete characterization of $k$-spectrally monomorphic Hermitian matrices and proves their spectral uniformity across different submatrix sizes.

Findings

Hermitian matrices with equal characteristic polynomials for all $k$-submatrices are characterized.

Such matrices are $l$-spectrally monomorphic for all $l$ in a specific range.

The paper establishes the spectral uniformity property for these 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 $\{1,\ldots,min\{k, n-k\}\}$.