General Polynomials over Division Algebras and Left Eigenvalues

TL;DR

This paper establishes a novel isomorphism linking polynomials over division algebras to group rings, enabling the definition of characteristic polynomials whose roots are the left eigenvalues, and demonstrates practical computation methods for matrices.

Contribution

It introduces a new isomorphism and defines characteristic polynomials over division algebras, facilitating the computation of left eigenvalues for matrices over such algebras.

Findings

Isomorphism between polynomial ring and group ring of free monoid

Definition of characteristic polynomial with roots as left eigenvalues

Method to find left eigenvalues of 4x4 matrices via degree 6 polynomial

Abstract

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $p^2$ variables. Using this isomorphism, we define the characteristic polynomial of a matrix over any division algebra, i.e. a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues. Plus, we show how the left eigenvalues of a $4 \times 4$ matrices over any division algebra can be found by solving a general polynomial equation of degree 6 over that algebra.