Abstract
Let A and B be two M-matrices, \(A^{-1}\) be the inverse of A, and \(\tau (B\circ A^{-1})\) be the minimum eigenvalue of the Hadamard product of B and \(A^{-1}\). Firstly, by using the theories of Schur complements, a lower bound of the main diagonal entries of \(A^{-1}\) is derived and used to present two types of lower bounds of \(\tau (B\circ A^{-1})\). Secondly, in order to obtain bigger lower bounds of \(\tau (B\circ A^{-1})\), two types of lower bounds of \(\tau (B\circ A^{-1})\) with non-negative parameters are constructed. Thirdly, by finding the optimal values of parameters, two preferable lower bounds of \(\tau (B\circ A^{-1})\) are yielded. Finally, numerical examples show the effectiveness of the new methods.
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Acknowledgements
The author is very grateful to the anonymous referees and Editor-in-Chief Prof. Rosihan M. Ali for their insightful comments and constructive suggestions, which considerably improve this manuscript. This work is supported by Guizhou Provincial Science and Technology Projects (Grant Nos. QKHJC-ZK[2021]YB013; QKHJC-ZK[2022]YB215), and Natural Science Research Project of Department of Education of Guizhou Province (Grant Nos. QJJ[2022]015; QJJ[2022]047).
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Zhao, J. Lower Bounds for the Minimum Eigenvalue of Hadamard Product of M-Matrices. Bull. Malays. Math. Sci. Soc. 46, 18 (2023). https://doi.org/10.1007/s40840-022-01432-8
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DOI: https://doi.org/10.1007/s40840-022-01432-8