Roots of composite polynomials—an application to root clustering

Abstract

In 1900, Stéphanos showed that given A ∈ C n × n with spectrum σ( A) = { λ i }, the eigenvalues of the composite matrix Φ=Σ p,qc pqA p⊗ A ̄ q are the n 2 values ∑ p, qc pqλ p i λ ̄ q j . This fact has been recently used in the theory of root clustering throug the characteristics polynomial of Φ. If a characteristic polynomial is given rather than A, we can use the companion matrix to generate the characteristic coefficients of Φ. In this paper we develop a direct theory for composite polynomials.