An analogue of the relationship between SVD and pseudoinverse over double-complex matrices

TL;DR

This paper extends the concept of the SVD and pseudoinverse to pairs of matrices over double numbers, introducing the Jordan SVD, which generalizes matrix decompositions for this algebraic setting.

Contribution

It introduces the Jordan SVD, a novel matrix decomposition for pairs of matrices over double numbers, building on previous work and generalizing classical SVD concepts.

Findings

Jordan SVD effectively generalizes classical SVD for matrix pairs

The new decomposition incorporates features of Jordan Normal Form

The approach treats pairs of matrices as single matrices over double numbers

Abstract

We present a generalisation of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called "Matrix decompositions over the double numbers". We adopt the idea presented in the same paper that a pair of matrices is actually a single matrix over the double number system.