Characterization of a family of generalized companion matrices

Abstract Matrices A of order n having entries in the field F ( x 1 , … , x n ) of rational functions over a field F and characteristic polynomial det ⁡ ( t I − A ) = t n + x 1 t n − 1 + ⋯ + x n − 1 t + x n are studied. It is known that such matrices are irreducible and have at least 2 n − 1 nonzero entries. Such matrices with exactly 2 n − 1 nonzero entries are called Ma–Zhan matrices. Conditions are given that imply that a Ma–Zhan matrix is similar via a monomial matrix to a generalized companion matrix (that is, a lower Hessenberg matrix with ones on its superdiagonal, and exactly one nonzero entry in each of its subdiagonals). Via the Ax–Grothendieck Theorem (respectively, its analog for the reals) these conditions are shown to hold for a family of matrices whose entries are complex (respectively, real) polynomials.