Majorization inequalities for valuations of eigenvalues using tropical algebra

Seminar
Speaker
Prof. Marianne Akian (INRIA Saclay--Ile-de-France and CMAP, Ecole Polytechnique)
Date
24/06/2015 - 11:30 - 10:30Add to Calendar 2015-06-24 10:30:00 2015-06-24 11:30:00 Majorization inequalities for valuations of eigenvalues using tropical algebra We consider a matrix with entries over the field of Puiseux series, equipped with its non-archimedean valuation (the leading exponent). We establish majorization inequalities relating the sequence of the valuations of the eigenvalues of a matrix with the tropical eigenvalues of its valuation matrix (the latter is obtained by taking the valuation entrywise). We also show that, generically in the leading coefficients of the Puiseux series, the precise asymptotics of eigenvalues, eigenvectors and condition numbers can be determined. For this, we apply diagonal scalings constructed from the dual variables of a parametric optimal assignment constructed from the valuation matrix. Next, we establish an archimedean analogue of the above inequalities, which applies to matrix polynomials with coefficients in the field of complex numbers, equipped with the modulus as its valuation. In particular, we obtain log-majorization inequalities for the eigenvalues which involve combinatorial constants depending on the pattern of the matrices. This talk covers joint works with Ravindra Bapat, Stéphane Gaubert, Andrea Marchesini, and Meisam Sharify. Third floor seminar room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room
Abstract

We consider a matrix with entries over the field of Puiseux series,
equipped with its non-archimedean valuation (the leading exponent).
We establish majorization inequalities relating the
sequence of the valuations of the eigenvalues of a matrix
with the tropical eigenvalues of its valuation matrix
(the latter is obtained by taking the valuation entrywise).
We also show that, generically in the leading coefficients of the
Puiseux series, the precise asymptotics of eigenvalues, eigenvectors
and condition numbers can be determined.
For this, we apply diagonal scalings constructed from
the dual variables of a parametric optimal assignment constructed from
the valuation matrix.

Next, we establish an archimedean analogue of the above inequalities,
which applies to matrix polynomials with coefficients in
the field of complex numbers, equipped with the modulus as its valuation.
In particular, we obtain log-majorization inequalities for the eigenvalues
which involve combinatorial constants depending on the pattern of the matrices.

This talk covers joint works with Ravindra Bapat, Stéphane Gaubert,
Andrea Marchesini, and Meisam Sharify.

Last Updated Date : 11/06/2015