Some results on the symmetric doubly stochastic inverse eigenvalue problem

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice‎, ‎East China Normal University‎, ‎Shanghai‎, ‎200241‎, ‎P‎. ‎R‎. ‎China.

2 Department of Mathematics‎, ‎Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice‎, ‎East China Normal University‎, ‎Shanghai‎, ‎200241‎, ‎P‎. ‎R‎. ‎China.

Abstract

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $\sigma=(1,\lambda_{2},\lambda_{3},\ldots,\lambda_{n})\in \mathbb{R}^{n}$ with $|\lambda_{i}|\leq 1,~i=1,2,\ldots,n$‎, ‎to be the spectrum of an $n\times n$ symmetric doubly stochastic matrix $A$‎.
‎If there exists an $n\times n$ symmetric doubly stochastic matrix $A$ with $\sigma$ as its spectrum‎, ‎then the list $\sigma$ is s.d.s‎. ‎realizable‎, ‎or such that $A$ s.d.s‎. ‎realizes $\sigma$‎. ‎In this paper‎, ‎we propose a new sufficient condition for the existence of the symmetric doubly stochastic matrices with prescribed spectrum‎. ‎Finally‎, ‎some results about how to construct new s.d.s‎. ‎realizable lists from the known lists are presented.

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Main Subjects


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