Polynomial eigenvalue bounds from companion matrix polynomials
A. Melman · Linear and Multilinear Algebra · 2018
We show how ℓ-ifications, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent results, namely, a generalization to matrix polynomials of the Cauchy radius and its enhancement, to produce eigenvalue bounds. In addition, we present a simplified derivation of ℓ-ifications for square matrices. We found that reasonably good bounds can be obtained by working with any companion matrix polynomial, and that these bounds were less computationally demanding for higher degree companion matrix polynomials than for such polynomials of degree one (linearizations).