Title: Digraph considerations for companion matrix configurations
Abstract: The Frobenius companion matrix has been the classic
tool for obtaining a matrix with a specified characteristic
polynomial. More recently, new companion matrices have
been described. Fiedler developed a product construction that
obtains a large class of companion matrices. Digraphs can be
used to characterize the sparse companion matrices and provide a
canonical Hessenberg form for the Fiedler companion matrices.
The new configurations lead to new bounds for roots
of polynomials. There is also opportunity to improve
on the condition number of the Frobenius companion matrix.