Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

Marianne Akian (1, 2, 3) , Stéphane Gaubert (1, 2, 3) , Meisam Sharify (4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MAXPLUS - Max-plus algebras and mathematics of decision
3 TROPICAL - TROPICAL
4 IPM - Institute for Research in Fundamental Sciences [Tehran]

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.

Domaines: Théorie spectrale [math.SP]; Analyse numérique [math.NA]

Marianne Akian, Stéphane Gaubert, Meisam Sharify. Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots. Linear Algebra and its Applications, 2017, 528, pp.394--435. ⟨10.1016/j.laa.2016.11.004⟩. ⟨hal-00881196⟩