A note on a relationship between the inverse eigenvalue problems for nonnegative and doubly stochastic matrices and some applications

Abstract

In this note, we establish some connection between the nonnegative inverse eigenvalue problem and that of doubly stochastic one. More precisely, we prove that if $(r; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative matrix A with Perron eigenvalue r, then there exists a least real number $k_A\geq -r$ such that $(r+\epsilon; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative generalized doubly stochastic matrix for all $\epsilon\geq k_A.$ As a consequence, any solutions for the nonnegative inverse eigenvalue problem will yield solutions to the doubly stochastic inverse eigenvalue problem. In addition, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Some related results are also discussed.Comment: 8 page