Linear Construction of Companion Matrices

This note is concerned with the following problem: For a given matrix A E C’Ix” and a vector a E C”, does there exist a mapping 2’ assigning to each manic polynomial f of degree n a vector X(f) E C” such that the matrix B := A a. X(f)’ is a companion matrix off, i.e., the characteristic polynomial of B is t l)“ftf’ The classes of suitable matrices A and vectors a are characterized, and some properties of B are described. The corresponding unique mapping ;% is determined by a system of linear equations. The cases of a triangular, bidiagonal, or diagonal matrix A are discussed explicitly, and many known companion matrices are obtained as particular cases. Then, Gershgorin’s theorem is applied, yielding error estimates for polynomial roots. Finally, the extension to block-companion matrices and an example of nonlinear construction are discussed.