A note on the real inverse spectral problem for doubly stochastic matrices

Abstract

The real (resp. symmetric) doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real n-tuple λ=(1,λ2,...,λn) to be the spectrum of an n×n (resp. symmetric) doubly stochastic matrix. If λi≤0 for all i=2,...,n and the sum of all the entries in λ is nonnegative, then we refer to such λ as a normalized Suleimanova spectrum.The purpose of this paper is to first fix an error in Theorem 9 of the recent paper Adeli et al. (2018) [1], after giving a counterexample. Secondly, we give a negative answer to a question posed in Johnson and Paparella (2016) [3] concerning the realizability of normalized Suleimanova spectra for the case when n is odd. Some sufficient conditions for a positive answer to this question are given.