Abstract
This note is concerned with the following problem: For a given matrix A∈ C n× n and a vector a∈ C n , does there exist a mapping K assigning to each monic polynomial f of degree n a vector K(f)∈ C n such that the matrix B ≔ A - a· K ( f) t is a companion matrix of f, i.e., the characteristic polynomial of B is (-1) n f? The classes of suitable matrices A and vectors a are characterized, and some properties of B are described. The corresponding unique mapping K is determined by a system of linear equations. The cases of a triangular, bidiagonal, or diagonal matrix A are discussed explicitl, 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.
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