The Cayley-Hamilton Theorem for Noncommutative Semirings

Abstract

The Cayley-Hamilton theorem (CHT) is a classic result in linear algebra over fields which states that a matrix satisfies its own characteristic polynomial. CHT has been extended from fields to commutative semirings by Rutherford in 1964. However, to the best of our knowledge, no result is known for noncommutative semirings. This is a serious limitation, as the class of regular languages, with finite automata as their recognizers, is a noncommutative idempotent semiring. In this paper we extend the CHT to noncommutative semirings. We also provide a simpler version of CHT for noncommutative idempotent semirings.