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.