Towards a Singular Value Decomposition and spectral theory for all rings

TL;DR

This paper introduces generalized definitions of SVD, spectral, and Jordan decompositions applicable to all rings, exploring their existence across various algebraic structures, including conjectures for Clifford algebras.

Contribution

It develops a unified framework for matrix decompositions over all rings, extending classical linear algebra concepts beyond fields and modules.

Findings

Decompositions exist for many rings, including some Clifford algebras.

Definitions are consistent with classical decompositions over fields.

Conjecture: these decompositions exist for a broad class of rings.

Abstract

We propose definitions of SVD, spectral decomposition (for self-adjoint matrices) and Jordan decomposition which make sense for all rings. For many rings, these decompositions can be shown to exist. For some specific rings, these decompositions are complicated to describe in full and prove the existence of. These decompositions have occurred piecemeal in the literature. We conjecture that they exist for many rings, including all Clifford algebras over the real numbers and complex numbers. The origin of this programme is not directly in module theory or linear algebra.