Generalisations of Singular Value Decomposition to dual-numbered matrices

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

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…