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 |lambda(i)| < 1 for i = 2, ..., n - 1 cannot be the spectrum of any n x 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 )is an element of [-1, 0, ..., 0, -1] 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, - 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 x 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.