Factorization of matrices of quaternions

We review known factorization results for quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, and QR factorization. We prove that there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature.Rather than work directly with matrices of quaternions, we work with complex matrices with a specific symmetry based on the dual operation. We discuss related results regarding complex matrices that are self-dual or symmetric, but perhaps not Hermitian.