Hermitizable, isospectral complex matrices or differential operators
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The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth–death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.
KeywordsReal spectrum symmetrizable Hermitizable isospectral matrix differential operator
MSC15A18 34L05 35P05 37A30
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The author thanks Ms. Yue-Shuang Li for her assistance. In particular, she has proved a partial answer for the existence of a positive solution to the h-transform in the tridiagonal case. The author also thanks Professor Tao Tang for sending to him (in March, 2018) a preprint of  where the problem on non-symmetric tridiagonal matrices arises. The author luckily found (within a week) a solution to the open question, it is now presented in Section 4. The careful corrections by the referees are also acknowledged. The results were presented in four times in January and March, 2018 at our seminar. The author obtained many helpful suggestions from our research group. The results have been reported at Beijing Normal U. (2018/3), Fudan U. (2018/4), Shanghai Jiaotong U. (2018/4), Fujian Normal U. (2018/4), USTC (2018/4), Jiangsu Normal U. (2018/4), Zhejiang U. (2018/5), Southwest Jiaotong U. (2018/7), Sichuan U. (2018/7), Workshop on Stochastic Analysis (FNU) (2018/4), and Workshop on Probability Theory its Applications (HUAS) (2018/7). The author acknowledges the following professors and their institutes for the invitation and financial support: Zeng-Hu Li and Zhong-Wei Tang, Da-Qian Li and Wei-Guo Gao, Dong Han, Huo-Nan Lin and Jian Wang, Jia-Yu Li, Tu-Sheng Zhang, Ying-Chao Xie, Gang Bao, Wei-Ping Li, An-Min Li, Ke-Ning Lu, Lian-Gang Peng, Wei-Nian Zhang, Xu Zhang, Xiang-Qun Yang and Xu-Yan Xiang. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11771046), the project from the Ministry of Education in China, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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