On the quotient matrix and its distinct eigenvalues

Abstract

Let $M$ be an $n\times n$ matrix partitioned into blocks according to a partition $P=\{C_1,\dots ,C_{\ell }\}$ of its index set, and let $Q$ be the corresponding equitable quotient. We study when $Q$ contains every distinct eigenvalue of $M$, so that the distinct spectral data of a large structured matrix can be recovered from a lower-order quotient. We first correct the distinction between a coarsest equitable partition and a minimum-order quotient that captures all distinct eigenvalues. We then establish an exact block-lifting construction with characteristic polynomial ${\chi }_M(x)={\chi }_Q(x){\prod }_i(x-{\theta }_i{)}^{|C_i|-1}$, derive multiplicity-design consequences, and give a necessary and sufficient eigenspace criterion for quotient capture. Further results quantify automatic capture through geometric multiplicity and describe exact spectral factors generated by twin cells in graph matrices. Applications to adjacency, weighted adjacency, Laplacian, signless Laplacian, distance, and distance-based Laplacian matrices are validated, including minimal one-cell refinements that recover eigenvalues hidden by collapsed twin classes.

arXiv comment40 pages, 1 figure, updated sevral results and fixed typos