Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

Abstract

Let [Formula: see text] be a real symmetric polynomial matrix of order [Formula: see text] and let [Formula: see text] be the largest eigenvalue function of the matrix [Formula: see text] We denote by [Formula: see text] the Clarke subdifferential of [Formula: see text] at [Formula: see text] In this paper, we first give the following nonsmooth version of Łojasiewicz gradient inequality for the function [Formula: see text] with an explicit exponent: For any [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that we have for all [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is a function introduced by D’Acunto and Kurdyka: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] Then we establish some local and global versions of Łojasiewicz inequalities which bound the distance function to the set [Formula: see text] by some exponents of the function [Formula: see text].