## 2013.25: Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

2013.25:
Marianne Akian, Stéphane Gaubert and Meisam Sharify
(2013)
*Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots.*

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## Abstract

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.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | Matrix polynomial, Tropical algebra, Majorization of eigenvalues, Tropical roots, Roots of polynomial, Bound of Pólya |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2013.25 |

Deposited By: | Dr Meisam Sharify |

Deposited On: | 15 May 2013 |

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