Non-archimedean valuations of eigenvalues of matrix polynomials

Marianne Akian; Ravindra Bapat; Stéphane Gaubert

doi:10.1016/j.laa.2016.02.036

Non-archimedean valuations of eigenvalues of matrix polynomials

Indian Statistical Institute (Delhi Centre)

Research output: Contribution to journal › Article › peer-review

Abstract

We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.

Original language	English
Pages (from-to)	592-627
Number of pages	36
Journal	Linear Algebra and Its Applications
Volume	498
DOIs	https://doi.org/10.1016/j.laa.2016.02.036
Publication status	Published - 1 Jun 2016

Keywords

Amoeba
Majorization
Max-plus algebra
Newton-Puiseux theorem
Perturbation theory
Spectral theory
Tropical semifield

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10.1016/j.laa.2016.02.036

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title = "Non-archimedean valuations of eigenvalues of matrix polynomials",

abstract = "We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.",

keywords = "Amoeba, Majorization, Max-plus algebra, Newton-Puiseux theorem, Perturbation theory, Spectral theory, Tropical semifield",

author = "Marianne Akian and Ravindra Bapat and St{\'e}phane Gaubert",

year = "2016",

month = jun,

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doi = "10.1016/j.laa.2016.02.036",

language = "English",

volume = "498",

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journal = "Linear Algebra and Its Applications",

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T1 - Non-archimedean valuations of eigenvalues of matrix polynomials

AU - Akian, Marianne

AU - Bapat, Ravindra

AU - Gaubert, Stéphane

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.

AB - We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials.

KW - Amoeba

KW - Majorization

KW - Max-plus algebra

KW - Newton-Puiseux theorem

KW - Perturbation theory

KW - Spectral theory

KW - Tropical semifield

U2 - 10.1016/j.laa.2016.02.036

DO - 10.1016/j.laa.2016.02.036

M3 - Article

AN - SCOPUS:84977954167

SN - 0024-3795

VL - 498

SP - 592

EP - 627

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Non-archimedean valuations of eigenvalues of matrix polynomials

Abstract

Keywords

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