Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices |
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Authors: | Jianzhou Liu Jicheng Li Zhuohong Huang |
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Affiliation: | a Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China b School of Mathematics, Xi’an Jiaotong University, Xi’an, Shanxi 710049, China |
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Abstract: | In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant. |
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Keywords: | Diagonally dominant matrix Schur complement H-matrix Diagonal-Schur complement |
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