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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: Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices | Chau and Lam Journal of Inequalities and Applications 2011 2011 18 http www.journalofinequalitiesandapplications.eom content 2011 1 18 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices Hoi Fung Chau1 2 and Yan Ting Lam1 3 Correspondence hfchau@hku.hk department of Physics University of Hong Kong Pokfulam Road Hong Kong Full list of author information is available at the end of the article SpringerOpen0 Abstract We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices-one by Childs et al. J. Mod. Phys. 47 155 2000 and the other one by Chau Quant. Inf. Comp. 11 721 2011 . Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators. Let Eig U denotes the set of all eigenvalues of a unitary matrix U. Interestingly one can give non-trivial information on Eig UV usually in the form of inequalities solely based on Eig U and Eig V . See for example Refs. 1 2 for comprehensive reviews of the field of spectral variation theory of matrices including Hermitian and normal ones. In this paper we give elementary proofs of two such inequalities. Let us begin by introducing a few notations first. Definition 1. Let U be a n-dimensional unitary matrix. Generalizing the conventions adopted in Ref. 2 we denote the arguments all arguments in this paper are in principal values of the eigenvalues of U arranged in descending and ascending orders by 0 U sand 0 U s respectively where the index j runs from 1 to n. That is to say 0j U e n n where 0J U e n n flnd lộỳ U is a normalized eigenvector of U with eigenvalue ei0 U . Moreover we write the eigenspace spanned by the eigenket Hj U by Hj U and the eigenspace corresponding to the .
