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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Row stochastic inverse eigenvalue problem | Shang-jun and Chang-qing Journal of Inequalities and Applications 2011 2011 24 http www.journalofinequalitiesandapplications.eom content 2011 1 24 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Row stochastic inverse eigenvalue problem Yang Shang-jun 1 and Xu Chang-qing2 Correspondence cqxurichard@163.com 2School of Sciences Zhejiang A F University Hangzhou 311300 China Full list of author information is available at the end of the article SpringerOpen0 Abstract In this paper we give sufficient conditions or realizability criteria for the existence of a row stochastic matrix with a given spectrum A 11 . 1n A1 u . u Am u Am 1 m 0 where Ak A-kx Ak2 - Akpk Akx k 2 -- Oke4 . Oke2Pk-1 iHpk Pk is an integer greater than 1 1k1 1k 0 1 11 rnk 0 k 1 . m Am 1 1m 1 fflm 1 -11 . 1n 11 mk 1k m1 1k k 2 . m 1. In the case when p1 . pm are all equal to 2 A becomes a list of 2m 1 real numbers for any positive integer m and our result gives sufficient conditions for a list of 2m 1 real numbers to be realizable by a row stochastic matrix. AMS classification 15A18. Keywords row stochastic matrices inverse eigenvalue problem row stochastic inverse eigenvalue problem 1 Introduction and preliminaries A list of complex numbers A is realizable by a matrix A if A is the spectrum of A. An n X n nonnegative matrix A amk is a row stochastic matrix if 52n x amk 1 m 1 . n. The row stochastic inverse eigenvalue problem is the problem of characterizing all possible spectrum of row stochastic matrices. Since row stochastic matrices are important in applications this kind of inverse eigenvalue problems should be interesting. There are also some papers that contribute to the study of the doubly stochastic inverse eigenvalue problem e.g. 1 2 and references therein . In this paper we give sufficient conditions for some lists of complex numbers including lists of 2m 1 real numbers to be realizable by a row stochastic matrix. We use A e CSr to denote the fact that the