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

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Abstract

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.

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Authors and Affiliations

  1. St. Petersburg Branch of the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023, Russian Federation

    Lilia Yu. Kolotilina

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  1. Lilia Yu. Kolotilina
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Correspondence to Lilia Yu. Kolotilina.

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Communicated by L. Cvetković.

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Kolotilina, L.Y. Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets. Numer Algor 42, 247–280 (2006). https://doi.org/10.1007/s11075-006-9041-7

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  • DOI: https://doi.org/10.1007/s11075-006-9041-7

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