Unitary canonical forms over Clifford algebras, and an observed unification of some real-matrix decompositions

We show that the spectral theorem – which we understand to be a statement every self-adjoint matrix admits certain type of canonical form under unitary similarity analogues over other ∗-algebras distinct from complex numbers. If these contain nilpotents, then it is shown there consistent way in many classic decompositions such as Singular Value Decomposition, Takagi decomposition, skew-Takagi and Jordan among others are immediate consequences these. producing relevant were subroutine some programming language, corresponding decomposition would 1-line invocation with no additional steps. also suggest by employing operator overloading numerical algorithm for computing diagonalization generalize immediately solving problems like SVD or Takagi. While algebras without nilpotents (like quaternions) allow similar unifying behaviour, they unify never obtained easily. In process doing this, develop theory Clifford Clp,q,0(R) Clp,q,1(R) where former admittedly quite easy. propose broad conjecture about theorems.