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

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.