Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets

详细信息    查看全文

• 作者：Lilia Yu. Kolotilina
• 关键词：Perron root ; nonnegative matrices ; spectral radius ; diagonal dominance ; singularity/nonsingularity ; eigenvalue inclusion sets ; directed graphs ; block ; partitioned matrices
• 刊名：Numerical Algorithms
• 出版年：2006
• 出版时间：July, 2006
• 年：2006
• 卷：42
• 期：3-4
• 页码：247-280
• 全文大小：657 KB
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

Using a unified approach based on the monotonicity property of the Perron root and its circuit extension, a series of exact two-sided bounds for the Perron root of a nonnegative matrix in terms of paths in the associated directed graph is obtained. A method for deriving the so-called mixed upper bounds is suggested. Based on the upper bounds for the Perron root, new diagonal dominance type conditions for matrices are introduced. The singularity/nonsingularity problem for matrices satisfying such conditions is analyzed, and the associated eigenvalue inclusion sets are presented. In particular, a bridge connecting Gerschgorin disks with Brualdi eigenvalue inclusion sets is found. Extensions to matrices partitioned into blocks are proposed.