Exploiting Equitable Partitions for Efficient Block Triangularization

TL;DR

This paper introduces a method to efficiently approximate block triangularization of matrices using equitable partitions, preserving key spectral properties and extending to weighted and rectangular matrices.

Contribution

It generalizes equitable partitions to weighted and rectangular matrices, providing an efficient unitary transformation for approximate block triangularization.

Findings

Exact in equitable case

Error bounds based on unitarily invariant norms

Preserves singular values and Hermiticity

Abstract

In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block by its row sum yields a smaller matrix whose multiset of eigenvalues is contained in the initial spectrum. We generalize this approach to weighted row sums and rectangular matrices and derive an efficient unitary transformation which approximately block triangularizes a matrix w.r.t. an arbitrary partition. Singular values and Hermiticity (if present) are preserved. The approximation is exact in the equitable case and the error can be bounded in terms of unitarily invariant matrix norms.