Universal Similarity Factorization Equalities over Generalized Clifford Algebras

Abstract

For any element a in a generalized 2ⁿ–dimensional Clifford algebra ${\fancyscript C}$ℓ _n (${\Bbb F}$) over an arbitrary field ${\Bbb F}$ of characteristic not equal to two, it is shown that there exits a universal invertible matrix P _n over ${\fancyscript C}$ℓ _n (${\Bbb F}$) such that $P^{{ - 1}}_{n} D_{a} P_{n} = \phi {\left( a \right)} \in F^{{2^{n} \times 2^{n} }}$, where ϕ(a) is a matrix representation of a over and D _a is a diagonal matrix consisting of a or its conjugate.