Matrix decompositions over the double numbers

TL;DR

This paper extends matrix decompositions to double numbers, unifying and simplifying linear algebra concepts over real, complex, and double number systems, and introduces a new Jordan SVD decomposition.

Contribution

It introduces the extension of matrix decompositions to double numbers and proposes the Jordan SVD, providing new insights and tools for linear algebra over this algebraic system.

Findings

Reduces LU to LDL decomposition over double numbers

Reduces eigendecomposition to singular value decomposition over double numbers

Proposes the new Jordan SVD decomposition

Abstract

We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For instance, we reduce the LU decomposition of real matrices to LDL decomposition of double matrices; we similarly reduce eigendecomposition of real matrices to singular value decomposition of double matrices. Notably, these are opposite to the usual reductions. This provides insight into linear algebra over the familiar real numbers and complex numbers. We also show that algorithms that are valid for complex matrices are often equally valid for double matrices. We finish by proposing a new matrix decomposition called the Jordan SVD, which we use to challenge a claim made in Yaglom's book Complex Numbers In Geometry concerning Linear Fractional Transformations over the double numbers.