Topological approach to diagonalization algorithms

TL;DR

This paper establishes a topological framework for diagonalization algorithms, showing their existence depends on the matrix's ability to be permuted into Hessenberg form, using advanced mathematical tools.

Contribution

It introduces a novel topological approach to characterize when diagonalization algorithms exist for certain sparse matrices.

Findings

Diagonalization algorithms exist if matrices can be permuted into Hessenberg form.

The proof employs Morse theory, Roberts' theorem, and toric topology.

Computer-based homological calculations support the theoretical results.

Abstract

In this paper we prove that there exists an asymptotical diagonalization algorithm for a class of sparse Hermitian (or real symmetric) matrices if and only if the matrices become Hessenberg matrices after some permutation of rows and columns. The proof is based on Morse theory, Roberts' theorem on indifference graphs, toric topology, and computer-based homological calculations.