On constructing orthogonal generalized doubly stochastic matrices

TL;DR

This paper introduces numerically stable algorithms for constructing orthogonal generalized doubly stochastic matrices, including solutions to inverse eigenvalue problems and matrices satisfying the Yang-Baxter equation, with demonstrated numerical effectiveness.

Contribution

It presents new stable methods for generating orthogonal g.d.s. matrices, solving inverse eigenvalue problems, and constructing matrices satisfying YBE, expanding computational tools in matrix theory.

Findings

Algorithms are numerically stable and effective.

Proposed methods successfully generate matrices with desired properties.

Numerical tests confirm the algorithms' usefulness.

Abstract

A real quadratic matrix is generalized doubly stochastic (g.d.s.) if all of its row sums and column sums equal one. We propose numerically stable methods for generating such matrices having possibly orthogonality property or/and satisfying Yang-Baxter equation (YBE). Additionally, an inverse eigenvalue problem for finding orthogonal generalized doubly stochastic matrices with prescribed eigenvalues is solved here. The tests performed in \textsl{MATLAB} illustrate our proposed algorithms and demonstrate their useful numerical properties.