On a numerical construction of doubly stochastic matrices with prescribed eigenvalues

TL;DR

This paper develops hybrid algorithms combining Dykstra's algorithm and alternating projections to construct doubly stochastic matrices with specified eigenvalues, proving convergence and demonstrating efficiency through numerical examples.

Contribution

It introduces new hybrid algorithms for the inverse eigenvalue problem of doubly stochastic matrices, with proven convergence and linear convergence rate.

Findings

Algorithms successfully construct doubly stochastic matrices with prescribed eigenvalues.

Proven convergence and linear convergence rate of the proposed algorithms.

Numerical examples demonstrate the efficiency of the methods.

Abstract

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method.