On symmetries of a matrix and its isospectral reduction

TL;DR

This paper explores how symmetries in the isospectral reduction of a diagonalizable matrix can be used to identify corresponding symmetries in the original matrix, enhancing understanding of eigenvalue problems.

Contribution

It demonstrates that symmetries found in the isospectral reduction can be systematically translated back to symmetries of the original matrix.

Findings

Symmetries in isospectral reductions imply symmetries in original matrices.

The approach aids in analyzing eigenvalue problems through reduced matrices.

Provides a method to construct original matrix symmetries from reduced forms.

Abstract

The analysis of diagonalizable matrices in terms of their so-called isospectral reduction represents a versatile approach to the underlying eigenvalue problem. Starting from a symmetry of the isospectral reduction, we show in the present work that it is possible to construct a corresponding symmetry of the original matrix.