Inverse Eigenvalue Problem of Cell Matrices

TL;DR

This paper investigates the inverse eigenvalue problem for cell matrices, showing how to reconstruct such matrices from spectral data and proving that their spectra are invariant under permutations of the defining vector.

Contribution

It introduces methods for reconstructing cell matrices from spectral data and establishes spectral invariance under permutations of the defining vector.

Findings

Reconstruction method for cell matrices from spectral data

Spectral invariance under permutation of vector elements

Characterization of eigenvalues of cell matrices

Abstract

In this paper, we consider the problem of reconstructing an $n \times n$ cell matrix $D(\vec{x})$ constructed from a vector $\vec{x} = (x_{1}, x_{2},\dots, x_{n})$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectrum of cell matrices $D(\vec{x})$ and $D(\pi(\vec{x}))$ are the same, for every permutation $\pi \in S_{n}$.