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Hermitizable, isospectral complex matrices or differential operators

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Abstract

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.

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References

  1. Berezanskii Yu M, Samoilenko V G. On the self-adjointness of differential operators with finitely or infinitely many variables, and evolution equations. Russian Math Surveys, 1981, 36(5): 1–62

    Article  Google Scholar 

  2. Brustentsev A G. (2004). Selfadjointness of elliptic differential operators in L 2(G), and correction potentials. Trans Moscow Math Soc, 2004, 65: 31–61

    Article  Google Scholar 

  3. Cao Z H. Eigenvalue Problem of Matrices. Shanghai: Shanghai Press of Sci & Tech, 1983 (in Chinese)

    Google Scholar 

  4. Chen M F. Exponential L 2-convergence and L 2-spectral gap for Markov processes. Acta Math Sin, New Ser, 1991, 7(1): 19–37

    Article  Google Scholar 

  5. Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004 (1st ed, 1992)

    Book  Google Scholar 

  6. Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005

    MATH  Google Scholar 

  7. Chen M F. Speed of stability for birth–death processes. Front Math China, 2010, 5(3): 379–515

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen M F. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen M F. Efficient initials for computing the maximal eigenpair. Front Math China, 2016, 11(6): 1379–1418 See also volume 4 in the middle of the author’s homepage: https://doi.org/math0.bnu.edu.cn/˜chenmf A package based on the paper is available on CRAN (by X. J. Mao): https://doi.org/cran.r-project.org/web/packages/EfficientMaxEigenpair/index.html A MatLab package is also available, see the author’s homepage above

    Google Scholar 

  10. Chen M F. The charming leading eigenpair. Adv Math (China), 2017, 46(4): 281–297

    MATH  Google Scholar 

  11. Chen M F. Global algorithms for maximal eigenpair. Front Math China, 2017, 12(5): 1023–1043

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen M F. Trilogy on computing maximal eigenpair. In: Yue W, Li Q L, Jin S, Ma Z, eds. Queueing Theory and Network Applications. QTNA 2017. Lecture Notes in Comput Sci, Vol 10591. Cham: Springer, 2017, 312–329

    MATH  Google Scholar 

  13. Chen M F. Mathematical topics motivated from statistical physics (I). Sci Sin Math, 2018 (to appear, in Chinese)

    Google Scholar 

  14. Chen M F. Mathematical topics motivated from statistical physics (II). Sci Sin Math, 2018 (to appear, in Chinese)

    Google Scholar 

  15. Chen M F, Zhang X. Isospectral operators. Commun Math Stat, 2014, 2: 17–32

    Article  MathSciNet  MATH  Google Scholar 

  16. Frolov A V, et al. Thomas algorithm, pointwise version. https://doi.org/algowiki-project.org/en/Thomas algorithm, pointwise version

  17. Hou Z T, Chen M F. Markov Processes and field theory (Abstract). Kuoxue Tongbao, 1980, 25(10): 807–811 (Complete version appeared in [23; pages 194–242])

    MATH  Google Scholar 

  18. Kato T. Remarks on the selfadjointness and related problems for differential operators. In: Knowles I W, Lewis R T, eds. Spectral Theory of Differential Operators. Amsterdam: North Holland, 1981, 253–266

    Book  MATH  Google Scholar 

  19. Kolmogorov A N. Zur Theorie der Markoffschen Ketten. Math Ann, 1936, 112: 155–160 (English translation: On the theory of Markov ehains. Article 21 in Selected Works of A. N. Kolmogorov, Vol II: Probability Theory and Mathematical Statistics, 182–187, edited by Shiryayev A N. Moscow: Nauka, 1986. Translated by Undquist G. Springer 1992)

    Article  MathSciNet  Google Scholar 

  20. Kolmogorov A N. Zur Umkehrbarkeit der statistischen Naturgesetze. Math Ann, 1937, 113: 766–772 (English translation: On the reversibility of the statistical laws of nature. Article 24 in Selected Works of A. N. Kolmogorov, Vol II: 209–215)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ludyk G. Quantum Mechanics in Matrix Form. Undergraduate Lecture Notes in Physics. Berlin: Springer, 2018

    Book  MATH  Google Scholar 

  22. Nino A, Munoz-Caro C, Reyes S. A concurrent object-oriented approach to the eigenproblem treatment in shared memory multicore environments. In: Lecture Notes in Comput Sci, Vol 6782. Cham: Springer, 2011, 630–642

    Google Scholar 

  23. Qian M, Hou Z T, eds. Reversible Markov Processes. Changsha: Hunan Sci & Tech Press, 1979 (in Chinese)

    Google Scholar 

  24. Schrödinger E. Über die Umkehrung der Naturgesetze. Sitzungsber Preuss, Akad Wiss, Phys-Math KI, 12 März, 1931, 144–153

  25. Shukuzawa O, Suzuki T, Yokota I. Real tridiagonalization of Hermitian matrices by modified Householder transformation. Proc Japan Acad Ser A, 1996, 72: 102–103

    Article  MathSciNet  MATH  Google Scholar 

  26. Tang T, Yang J. Computing the maximal eigenpairs of large size tridiagonal matrices with O(1) number of iterations. Numer Math Theory Methods Appl, 2018, 11(4): 877–894

    MathSciNet  Google Scholar 

  27. Varga R S. Geršgorin and His Circles. Berlin: Springer, 2004

    Book  MATH  Google Scholar 

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Acknowledgements

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 [26] 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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  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing, 100875, China

    Mu-Fa Chen

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  1. Mu-Fa Chen
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Chen, MF. Hermitizable, isospectral complex matrices or differential operators. Front. Math. China 13, 1267–1311 (2018). https://doi.org/10.1007/s11464-018-0716-x

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