# 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.

- Last modified: 11/06/2015