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On the solvability and Picard-type method for absolute value matrix equations

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Abstract

In this paper, we first present new sufficient conditions, necessary and sufficient conditions for the unique solvability of the absolute value matrix equation \(AX+C|X|=E\). New solvability conditions for the unique solvability of the generalized absolute value matrix equation \(AXB+C|X|D=E\) are also given. In which, \(A,~B,~C,~D,~E\in {\mathbb {R}}^{n\times n}\) are given, \(X\in {\mathbb {R}}^{n\times n}\) is an unknown matrix and \(|X|=(|x_{ij}|)\). Based on the new solvability conditions, we then study the convergence of Picard-type method for solving \(AX+C|X|=E\) and \(AXB+C|X|D=E\), and an example is given to show the theoretical results.

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Acknowledgements

We would like to thank the Editor and the anonymous referees for their valuable comments and suggestions that have led to a much improved presentation of the paper. This work was supported by National Natural Science Foundation of China (Nos. 11861059, 61967014) and the Foundation for Distinguished Young Scholars of Gansu Province (Grant No. 20JR5RA540).

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  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Wen-Ling Tang & Shu-Xin Miao

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  1. Wen-Ling Tang
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  2. Shu-Xin Miao
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Correspondence to Shu-Xin Miao.

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Communicated by Jinyun Yuan.

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Tang, WL., Miao, SX. On the solvability and Picard-type method for absolute value matrix equations. Comp. Appl. Math. 41, 78 (2022). https://doi.org/10.1007/s40314-022-01782-w

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  • DOI: https://doi.org/10.1007/s40314-022-01782-w

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