Standard Polynomial Equations over Division Algebras

2015/08/01 by Adam Chapman, Chapman, Adam, Casey Machen +1 · 4 citations
#11R52 #15B33 #FOS: Mathematics #Rings and Algebras (math.RA) #primary 16K20 #secondary 15A18

paper · doi:10.48550/arxiv.1508.00173

Abstract

Given a central division algebra D of degree d over a field F, we associate to any standard polynomial ϕ(z)=zⁿ+c_n-1 z^n-1+…+c₀ over D a "companion polynomial" Φ(z) of degree n d with coefficients in F whose roots are exactly the conjugacy classes of the roots of ϕ(z). We explain how in case D is a quaternion algebra, all the roots of ϕ(z) can be recovered from the roots of Φ(z). On the way, we also generalize certain theorems that were known for ℍ to any division algebra, such as the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial, and the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix.

Cited by

• General Polynomials and Eigenvalues Over Cayley--Dickson Algebras
• Observer Design over Hypercomplex Quaternions
• Quaternionic Pole Placement via Companion Forms and the Ackermann Formula
• Recurrence relations over division algebras

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