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On Hermitian positive definite solutions of the nonlinear matrix equation \(X-A^{*}e^{X}A=I\)

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Abstract

In this paper, the nonlinear matrix equation \(X-A^{*}e^{X}A=I\) is studied. Some sufficient and necessary conditions for the existence of the Hermitian positive definite solution are given. Then the distribution of the solution is discussed. At last, the basic fixed point iterative method for obtaining the unique positive definite solution is constructed.

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Acknowledgments

The work was supported by the National Natural Science Foundation of China (11071141), and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02).

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Authors and Affiliations

  1. Department of Mathematics, Heze University, Heze, 274000, Shandong, People’s Republic of China

    Dongjie Gao

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  1. Dongjie Gao
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Correspondence to Dongjie Gao.

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Gao, D. On Hermitian positive definite solutions of the nonlinear matrix equation \(X-A^{*}e^{X}A=I\) . J. Appl. Math. Comput. 50, 109–116 (2016). https://doi.org/10.1007/s12190-014-0861-7

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  • DOI: https://doi.org/10.1007/s12190-014-0861-7

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