Concentration of the mixed discriminant of well-conditioned matrices

TL;DR

This paper establishes bounds on the mixed discriminant of well-conditioned matrices and provides a polynomial-time approximation algorithm for it, advancing computational methods in matrix analysis.

Contribution

It proves bounds on the mixed discriminant for alpha-conditioned doubly stochastic matrices and introduces an efficient approximation algorithm.

Findings

Mixed discriminant is bounded by n^{O(1)} e^{-n} for alpha-conditioned matrices.

Provides a polynomial-time algorithm for approximating the mixed discriminant.

Results hold for fixed alpha > 1, independent of matrix size.

Abstract

We call an n-tuple Q_1, ..., Q_n of positive definite nxn matrices alpha-conditioned for some alpha > 1 if the ratio of the largest among the eigenvalues of Q_1, ..., Q_n to the smallest among the eigenvalues of Q_1, ..., Q_n does not exceed alpha. An n-tuple is called doubly stochastic if the sum of Q_i is the identity matrix and the trace of each Q_i is 1. We prove that for any fixed alpha > 1 the mixed discriminant of an alpha-conditioned doubly stochastic n-tuple is n^{O(1)} e^{-n}. As a corollary, for any alpha > 1 fixed in advance, we obtain a polynomial time algorithm approximating the mixed discriminant of an alpha-conditioned n-tuple within a polynomial in n factor.