Unitary canonical forms over Clifford algebras, and an observed unification of some real-matrix decompositions

TL;DR

This paper extends the spectral theorem to various $*$-algebras, unifying many classical matrix decompositions like SVD and Jordan forms through Clifford algebra spectral theory, with implications for programming and numerical algorithms.

Contribution

It introduces a unified framework for matrix decompositions via spectral theorems over Clifford algebras, connecting algebraic structures to classical decompositions.

Findings

Spectral theorems over Clifford algebras unify classical decompositions.

Matrix decompositions can be seen as canonical forms over different $*$-algebras.

Operator overloading can generalize algorithms for matrix diagonalization.

Abstract

We show that the spectral theorem -- which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity -- admits analogues over other $*$-algebras distinct from the complex numbers. If these $*$-algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions -- such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others -- are immediate consequences of these. If producing the relevant canonical form of a self-adjoint matrix were a subroutine in some programming language, then the corresponding classic matrix decomposition would be a 1-line invocation with no additional steps. We also suggest that by employing operator overloading in a programming language, a numerical algorithm for computing a unitary diagonalisation of a complex self-adjoint matrix would generalise immediately to solving problems like SVD or Takagi. While algebras without nilpotents (like the quaternions) allow for similar unifying behaviour, the classic matrix decompositions which they unify are never obtained as easily. In the process of doing this, we develop some spectral theory over Clifford algebras of the form $\cl_{p,q,0}(\mathbb R)$ and $\cl_{p,q,1}(\mathbb R)$ where the former is admittedly quite easy. We propose a broad conjecture about spectral theorems.