Generalizations of singular value decomposition to dual-numbered matrices

We present a generalisation of Singular Value Decomposition to arbitrary square dual-numbered matrices. This generalises a classification of invertible $2 \times 2$ dual matrices found in Yaglom's 1968 book \emph{Complex numbers in geometry}. Unlike the Singular Value Decomposition over real numbers or over complex numbers, our generalisation replaces diagonal matrices with block diagonal matrices where the blocks may be $1 \times 1$ or $2 \times 2$.