Relating the spectrum of a matrix and a principal submatrix using adjugates and Schur complements

TL;DR

This paper explores relationships between the spectra of a matrix and its principal submatrix using adjugates and Schur complements, aiming to simplify eigenvalue problems especially in graph theory.

Contribution

It introduces new relations based on annihilating polynomials to connect determinants and spectra of matrices and their submatrices.

Findings

Provides formulas linking determinants of matrices and submatrices.

Facilitates size reduction in eigenvalue problems for matrices with low-degree minimal polynomials.

Applies to spectral analysis of perturbed strongly regular graphs.

Abstract

Let $\mathcal{M}$ be a square matrix over a commutative ring and let $\mathcal{A}$ be a principal submatrix. We give relations between the determinants of $\mathcal{M}$ and $\mathcal{A}$ based on an annihilating polynomial for one of them. The intended application is the size reduction of complex latent root problems, especially the reduction of ordinary eigenvalue problems if a matrix or its principal submatrix have a low degree minimal polynomial. An example is the spectrum of vertex perturbed strongly regular graphs.