On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices

Abstract: Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P={C_{1},\dots,C_{\ell}}$ of index set ${1,\dots,n}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k \leq n-1$, where $(ij)$-th entry is the average row sum (or column sum) of the corresponding block $b_{ij}$ in $M$. The partition $P$ is said to be \emph{equitable} if row sum of each block $b_{ij}$ is constant. In this case, the matrix $Q$ is referred to as the \emph{equitable quotient matrix} of $M$, and the spectrum of $Q$ is the subset of the spectrum of parent matrix $M$. We characterize some classes of matrices such that their equitable quotient matrix $Q$ contains all the distinct eigenvalues of $M$, thereby information can be obtained form the smallest matrix $Q$ without actually analyzing the parent matrix $M.$ We present necessary and the sufficient conditions for distinct eigenvalue of $M$ contained in the spectrum of of $Q$ in terms of eigenspaces. We end up article with some applications, where distinct eigenvalues of a parent matrix can be completely encoded by quotient matrix.

• The paper presents necessary and sufficient criteria for when every distinct eigenvalue of a matrix appears in its equitable quotient matrix.
• It leverages block-structured analysis and equitable partition theory to construct quotient matrices that fully capture spectral data.
• Results extend to graph matrices, offering practical insights for spectral graph theory and scalable eigenvalue reduction.

Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices

Introduction

This work undertakes a systematic investigation into conditions under which the set of distinct eigenvalues (the spectrum) of a square matrix $M\in\mathcal{M}_n(\mathbb{R})$ coincides with those of a suitably constructed equitable quotient matrix $Q$. By leveraging block-structured matrix analysis and the theory of equitable partitions, the author establishes crisp criteria—necessary and sufficient—for when each distinct eigenvalue of $M$ is present in the spectrum of $Q$, and provides rich characterizations in terms of eigenspaces and the intersection with canonical invariant subspaces induced by the partition.

Equitable Partitions, Quotient Matrices, and Spectral Interlacing

Given a partition $P = \{C_1,\ldots,C_\ell\}$ of the index set $\{1,\ldots,n\}$, any $M\in\mathcal{M}_n(\mathbb{R})$ may be decomposed into a block form. The quotient matrix $Q$ associated with $P$ has as its $(i,j)$ entry the average row sum (of block $Q$0) in $Q$1. When each block has constant row-sums, the partition is equitable, and standard results guarantee that $Q$2. The classical eigenvalue interlacing theorem ensures that for non-equitable partitions, the eigenvalues of $Q$3 interlace those of $Q$4 (cf. Haemers).

Classes of Matrices Whose Quotient Matrices Encode All Distinct Eigenvalues

The paper gives strong characterizations for when all distinct eigenvalues of $Q$5 can be recovered from $Q$6, addressing and extending a question originally posed by Atik. The key theoretical advance is the identification of minimal equitable partitions for which the equitable quotient matrix $Q$7 contains all distinct eigenvalues of $Q$8, not merely a subset. Theorem \ref{thm:equitable-all-eigs-general} articulates necessary and sufficient conditions in terms of the intersection between each eigenspace of $Q$9 and the subspace $M$0 of vectors that are constant on each partition cell, i.e.,

$M$1

Explicit constructions are presented: for example, for the family

$M$2

with the partition $M$3, the two distinct eigenvalues of $M$4 are both present in the spectrum of the $M$5 quotient matrix $M$6.

Constructive results are provided for matrices with two eigenvalues (one of high multiplicity), as well as matrices with three or more distinct eigenvalues, for which the spectral information can be retrieved using quotient matrices given by block-structured canonical forms parametrized by partition data and eigenvalue assignments.

Application to Graph Matrices

The transfer of these spectral results to structured matrices arising in algebraic graph theory is nontrivial. The paper considers adjacency, Laplacian, signless Laplacian, and distance-based matrices. For certain non-regular graphs and specific families such as those created by attaching pendant vertices to a $M$7, the quotient matrix with an appropriate equitable partition encodes all distinct eigenvalues of the adjacency matrix, as detailed in Proposition \ref{rare graph unicyclic}. Extensions of this idea are achieved for general weighted adjacency matrices by parametric selection of the weight function $M$8.

However, for Laplacian-type matrices, the smallest possible equitable quotient may not contain all distinct eigenvalues—due to inherent row-sum constraints—necessitating further enlargement of the partition. The precise interaction between structural duplications (e.g., vertices with identical neighborhoods or symmetries in block structure) and spectral multiplicities is clarified, and an explicit (if not exhaustive) recipe for generating partitions capturing all spectrum points is given.

Generalizations to complete bipartite graphs and split graphs are provided, where iterative refinement of the equitable partition (e.g., splitting vertex classes) recovers all distinct eigenvalues in the quotient, even for highly nonregular cases.

Numerical Results and Theoretical Implications

For the presented matrix classes and graph families, explicit calculations confirm that the equitable quotient matrix spectrum matches the set of distinct eigenvalues of the parent matrix. For instance, for the family above, the spectrum of $M$9 is $Q$0, and the corresponding $Q$1 quotient matrix has eigenvalues exactly $Q$2.

These results show that, at least for certain block structures and network motifs, spectral data reduction via equitable partitioning is complete with respect to distinct eigenvalues. However, for arbitrary matrices or those associated with Laplacian-like operators, such completeness may require partition refinement.

Contradictory cases—where a distinct eigenvalue is present in $Q$3 but not in any quotient matrix with small partition—are constructed for Laplacian and signless Laplacian matrices, showing strict limitations on the generality of the phenomenon.

Future Directions

This work leaves open a full classification of matrices (and graph classes) for which some (possibly not minimal) equitable quotient matrix encodes all distinct eigenvalues, as well as the extension to non-equitable or almost equitable partitions. Further research may address stability under perturbation, computational aspects for large-scale graphs, and connections with algebraic combinatorics (e.g., association schemes). The bridge to algorithmic spectral graph theory is direct: for very large, highly symmetric graphs, the possibility of using a much smaller quotient matrix to analyze the spectrum suggests scalable heuristics for eigenvalue computation and for the analysis of dynamics (e.g., consensus, diffusion) driven by those spectra.

Conclusion

This paper offers a rigorous framework for understanding when all distinct eigenvalues of a matrix can be recovered from an equitable quotient matrix, with sharp characterizations in terms of eigenspace geometry and partition structure. Notable contributions include necessary and sufficient conditions for spectral completeness, explicit constructions for broad matrix families and graph classes, and a detailed exploration of the limitations for Laplacian-type matrices. The presented theory not only extends classical results but also provides practical tools for spectral analysis of large block-structured or symmetric matrices via dimensional reduction.

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Reference:

"On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices" (2604.03194)