A new aspect of Hankel matrices via Krylov matrix

An investigation is made of the Krylov matrices associated to the companion matrix C of a monic polynomial p(x) of degree n. In this paper, we propose a new approach to the study of Hankel matrices as Krylov matrices of CT. Firstly we focus on algebraic structures of Krylov matrices of C over a field F. We then discuss an equivalent condition for the Hankel matrix to be a Krylov matrix of CT by the notion of compatibility. As a result, we derive new determinant formulas for such Hankel matrices in terms of eigenvalues and eigenvectors of C respectively.