Characteristic polynomials in Clifford algebras and in more general algebras (Q2416607)

This paper is written for researchers in matrix theory and algebra. It relates the Faddeev-Leverrier algorithm for characteristic polynomial computations of matrices, descent theory (relating matrix and Azumaya algebras) and applications to Clifford algebras. The introduction reviews the notions of a vector space over a field, a Clifford algebra over fields, its center, characteristic polynomials, determinants, traces, the Faddeev-Leverrier algorithm and the notion of Azumaya algebra. Section 2 is on polynomials, the first theorem (Theorem 2.1) being on ring morphisms applied to ring polynomials in one variable. Lemma 2.2 extends this to products of polynomials. The discussion extends it further to polynomial functions. The section concludes with the Newton identities for symmetric polynomials. Section 3 discusses Azumaya algebras, useful for Clifford algebras, because some of the latter have centers that are not fields. Concepts of modules over (commutative) rings are recalled. Azumaya algebras are algebras \(A\) over a ring \(\mathbb{R}\), if \(A\) is finitely generated and a faithful projective module over \(\mathbb{R}\), and the canonical morphism being bijective (for details see: [\textit{J. Helmstetter} and \textit{A. Micali}, Quadratic mappings and Clifford algebras. Basel: Birkhäuser (2008; Zbl 1144.15025), Section 3.5]). Section 4 reviews characteristic polynomials for matrices, systems of linear equations, the Cayley-Hamilton theorem (including a self-contained proof) and the background of the Faddeev-Leverrier algorithm for matrices. It also provides an interesting historical background. Section 5 discusses cases of descent, when \(\mathbb{R}\) is an infinite field, a finite field, and \(\mathbb{R}\) being the square of some field \(\mathbb{K}\). Section 6 gives applications to Clifford algebras, including their even subalgebras. Theorem 6.1 is on characteristic polynomials and adjoint elements for even subalgebras of Clifford algebras over a vector space of odd dimension. Lemma 6.2 is on products of polynomials in Clifford algebra variables with coefficients from the fields. Theorem 6.3 is analogous to Theorem 6.1 but for Clifford algebras over a vector space of even dimension. Lemma 6.4 is on the subspace spanned by commutators of elements of a Clifford algebra. Finally, the section concludes by discussing the application of the Faddeev-Leverrier algorithm to characteristic polynomials of Clifford algebras over vector spaces of even and odd dimension. Section 7 gives two explicitly computed examples of characteristic polynomials, adjoint elements and inverses in Clifford algebra \(\mathrm{Cl}\)(5,1) for an element with scalar, vector, bivector and trivector parts, and a second example for an element from the even subalgebra of \(\mathrm{Cl}\)(5,1).