Symmetric doubly stochastic inverse eigenvalue problem for odd sizes

The symmetric doubly stochastic inverse eigenvalue problem seeks to determine the necessary and sufficient conditions for a real list of eigenvalues to be realized by a symmetric doubly stochastic matrix. Nader et al. (2019) [15], established that for odd integers n a list of the form $\sigma =(1,{\lambda }_2,{\lambda }_3,...,{\lambda }_{{}_{n-1}},-1)$ with $\left|{\lambda }_i\right|<1$ for $i=2,...,n-1$ cannot be the spectrum of any $n\times n$ doubly stochastic matrix. This implies that the list $\sigma =(1,0,...,0,-1)$ is also unrealizable.

This paper extends these findings by proving that for odd n and ${\lambda }_{{}_n}\in [-1,-\frac{n-1}{n})$, the list $(1,0,...,0,{\lambda }_{{}_n})$ cannot be the spectrum of a symmetric doubly stochastic matrix. We demonstrate that for odd n the list $\sigma =(1,0,...,0,-\frac{n-1}{n})$, is indeed realizable as the spectrum of a symmetric doubly stochastic matrix.

Furthermore, we utilize our methodology to derive new sufficient conditions for the existence of $n\times n$ symmetric doubly stochastic matrices with a prescribed list of eigenvalues. This leads to a condition for the existence of symmetric doubly stochastic matrices with a normalized Suleimanova spectrum. The paper concludes with additional partial results and illustrative examples.