Matrix decompositions over the double numbers

Abstract: We take matrix decompositions that are usually applied to matrices over the real numbers or complex numbers, and extend them to matrices over an algebra called the double numbers. In doing so, we unify some matrix decompositions: For instance, we reduce the LU decomposition of real matrices to LDL decomposition of double matrices; we similarly reduce eigendecomposition of real matrices to singular value decomposition of double matrices. Notably, these are opposite to the usual reductions. This provides insight into linear algebra over the familiar real numbers and complex numbers. We also show that algorithms that are valid for complex matrices are often equally valid for double matrices. We finish by proposing a new matrix decomposition called the Jordan SVD, which we use to challenge a claim made in Yaglom's book Complex Numbers In Geometry concerning Linear Fractional Transformations over the double numbers.