On functions of companion matrices

Presented in this paper are some new properties of a function f(C) of a companion matrix C, including (1) a representation of any entry of f(C) as a divided difference of f(λ) times a polynomial, (2) a similarity decomposition of f(C) generalizing that based on the Jordan form of C, and (3) characterization (construction) of all matrices that transform f(C) (by similarity) to companion form. The connection between functions of general and companion matrices is also dealt with, and a pair of dual relations is established.