Generalisations of Singular Value Decomposition to dual-numbered matrices

TL;DR

This paper extends Singular Value Decomposition to dual-numbered matrices, providing two types of SVD and proofs of their existence, motivated by applications in geometry and mechanics.

Contribution

It introduces two novel generalisations of SVD for dual-numbered matrices and proves their existence, expanding the mathematical tools available for geometry and mechanics applications.

Findings

Both types of SVD exist for all dual-numbered matrices.

The generalisations are motivated by practical applications in geometry and mechanics.

The paper provides formal proofs of the existence of these SVDs.

Abstract

We present two generalisations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. We prove that every dual-numbered matrix has both types of SVD. Both of our generalisations are motivated by applications, either to geometry or to mechanics.