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Positive Definite Solution of a Class of Matrix Equations

Published in Innovation (Volume 4, Issue 2)
Received: 12 November 2023     Accepted: 29 November 2023     Published: 5 December 2023
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Abstract

The sources and application fields of nonlinear matrix equations are quite extensive, including control theory, statistics, dynamic programming, etc. A large number of problems can be transformed into solving matrix equations. In this paper we establish some necessary and sufficient conditions for the existence of positive definite solutions to the nonlinear matrix equation ATXA = ηX. The existence of positive definite solutions to corresponding inequalities were discussed too. In addition, some examples are presented to illustrate the main results of this paper.

Published in Innovation (Volume 4, Issue 2)
DOI 10.11648/j.innov.20230402.12
Page(s) 25-28
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Positive Definite Solution, Matrix Equations, Nonlinear Equations

References
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[2] Gohberg I, Lancaster P, Rodman L. Matrix Polynomials. Philadelphia: Academic Press, 1982.
[3] Higham N J, Kim H. Numerical analysis of a quadratic matrix equation. IMA Journal of numerical Analysis, 2000, 20: 499-519.
[4] Lancaster P, Rodman L. Algebraic Riccati Equations. Oxford: Oxford Science Publishers, 1995.
[5] Eisenfeld J. Operator equations and nonlinear eigenparameter problems. Journal Functional, 1973, 12: 475-490.
[6] Reurings M C B. Symmetric Matrix Equations. Netherland: The Netherland: Universal Press, 2003.
[7] Lin W W. An SDR algorithm for the solution of the generalized algebraic Riccati equation. IEEE Transactions on Automatic Control, 1989, 34 (8): 875- 879.
[8] Kleinman D L. On an iterative technique for Riccati equation computations. IEEE Transactions on Automatic Control, 1968, 13: 114- 115.
[9] Kalmanre. Contributions to the theory of optimal control. Bulletin Society Mathematics of Mexico, 1961, 5: 102- 119.
[10] Guo C H, Lancaster P. Analysis and modification of Newton’s method for algebraic Riccati equations. Mathematics of Computation, 1998, 67 (223): 1089- 1105.
[11] Jung C, Kim H M, Lim Y. On the solution of the nonliear matrix equation Xn = f(x). Linear Algebra and Its Applications, 2009, 430: 2042-2052.
[12] Kim H. Numerical methods for Solving a quadratic Matrix Equation. Manchester : Manchester Press, 2000.
[13] Zhang Y H. On Hermitian positive definite solutions of matrix equation A − A∗X−2A = I. Comput. Math., 2005 (23): 408-418.
[14] Hasanov V I. Positive definite solutions of the matrix equations X ± A∗X−qA = Q. Linear Algebra Appl, 2005, 404: 166-182.
[15] Ivanov I G. On positive definite solutions of the family of matrix equations X + A∗X−nA = Q. J. Comput. Appl. Math., 2006, 193 (1): 277-301.
[16] Duan X F, Liao A P. On the existence of Hermitian positive definite solutions of the matrix equation Xs± A∗X−1A = Q. Linearalgebraanditsapplications, 2008, 429 (4): 673-687.
[17] Liu W. Hermitian Positive Definite Solutions of the Matrix Equation X + A∗X−qA = Q (q ≥ 1). Journal of Mathematical Research with Applications, 2009, 29: 831-838.
[18] Guo B L, Chen Z. Real Similar to the Standard Form of Orthogonal Matrix, Journal of Yangtze University, 2014, 11: 11-13.
[19] Wu L. The Re-positive definite solutions to the matrix inverse problem AX=B, Linear Algebra and Its Applications, 1992, 174.
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    Yang, R. (2023). Positive Definite Solution of a Class of Matrix Equations. Innovation, 4(2), 25-28. https://doi.org/10.11648/j.innov.20230402.12

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    Yang, R. Positive Definite Solution of a Class of Matrix Equations. Innovation. 2023, 4(2), 25-28. doi: 10.11648/j.innov.20230402.12

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    Yang R. Positive Definite Solution of a Class of Matrix Equations. Innovation. 2023;4(2):25-28. doi: 10.11648/j.innov.20230402.12

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  • @article{10.11648/j.innov.20230402.12,
      author = {Ran Yang},
      title = {Positive Definite Solution of a Class of Matrix Equations},
      journal = {Innovation},
      volume = {4},
      number = {2},
      pages = {25-28},
      doi = {10.11648/j.innov.20230402.12},
      url = {https://doi.org/10.11648/j.innov.20230402.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.innov.20230402.12},
      abstract = {The sources and application fields of nonlinear matrix equations are quite extensive, including control theory, statistics, dynamic programming, etc. A large number of problems can be transformed into solving matrix equations. In this paper we establish some necessary and sufficient conditions for the existence of positive definite solutions to the nonlinear matrix equation ATXA = ηX. The existence of positive definite solutions to corresponding inequalities were discussed too. In addition, some examples are presented to illustrate the main results of this paper.
    },
     year = {2023}
    }
    

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    AB  - The sources and application fields of nonlinear matrix equations are quite extensive, including control theory, statistics, dynamic programming, etc. A large number of problems can be transformed into solving matrix equations. In this paper we establish some necessary and sufficient conditions for the existence of positive definite solutions to the nonlinear matrix equation ATXA = ηX. The existence of positive definite solutions to corresponding inequalities were discussed too. In addition, some examples are presented to illustrate the main results of this paper.
    
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Author Information
  • School of Mathematics and Statistics, Shandong Normal University, Jinan, P. R. China

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