Relating the spectrum of a matrix and a principal submatrix using adjugates and Schur complements
Provides a theoretical tool for reducing computational complexity in eigenvalue problems for matrices with low-degree minimal polynomials, relevant to spectral graph theory and linear algebra.
The paper presents relations between the determinants of a matrix and its principal submatrix using annihilating polynomials, enabling size reduction of eigenvalue problems when the matrix or submatrix has a low-degree minimal polynomial. An example application is given for vertex-perturbed strongly regular graphs.
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.