On bounds of extremal eigenvalues of irreducible and m-reducible matrices
On bounds of extremal eigenvalues of irreducible and m-reducible matrices
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DOI：
10.1016/j.laa.2004.12.004
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发表时间：
2005-06
影响因子：
1.1
通讯作者：
C. Wu
中科院分区：
文献类型：
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作者：
C. Wu
对于一个复矩阵A，著名的Lévy-Desplanques定理指出，A是非奇异的，如果|AII|>∑j i| AIJ|为了所有我。关于特征值局部化的等价Gershgorin定理意味着A的特征值λ必须满足|λ|中国（|AII|-∑j i| AIJ|). Taussky通过证明A是非奇异的，如果A是不可约的，|AII| ⩾∑j≠i| AIJ|对于至少一个i，不等式严格。本文的一个目标是给出这种情况下的<$λ <$的下界。给出了依赖于图A的直径和代数连通度的界。我们还通过引入m-可约性的概念来研究可约矩阵的界。特别是，我们给出了可约矩阵的界限，这取决于图的强连通分量的代数连通性和它们之间的边数。这些界也适用于可约随机矩阵和有向图的Laplacian矩阵的次优势特征值的界。
For a complex matrix A, the well-known Lévy–Desplanques theorem states that A is nonsingular if |Aii|>∑j≠i|Aij| for all i. The equivalent Gershgorin theorem on the localization of eigenvalues implies that the eigenvalues λ of A must satisfy |λ|⩾mini(|Aii|-∑j≠i|Aij|). Taussky extended this by showing that A is nonsingular if A is irreducible and |Aii|⩾∑j≠i|Aij| with the inequality strict for at least one i. A goal of this paper is to give lower bounds on ∣λ∣ for this case as well. We give bounds which depend on the diameter and the algebraic connectivity of the graph of A. We also study bounds for reducible matrices by introducing the notion of m-reducibility. In particular, we give bounds for reducible matrices which depend on the algebraic connectivity of the strongly connected components of the graph and the number of edges between them. These bounds are also applicable to bound the subdominant eigenvalues of reducible stochastic matrices and Laplacian matrices of directed graphs.