  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.
