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Symmetric matrices associated with a nonnegative matrix

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Abstract

For ann-by-n nonnegative matrixP, we consider the entrywise harmonic meanH, geometric meanG, and arithmetic meanA, ofP and PT. Simple proofs are given for the inequalities ρ(H)≤ρ(G)≤ p(P)≤ ρ(A), and attention is focused upon characterization of the case of equality in each of these six inequalities. In caseP is irreducible, ρ(G)=p(P) exactly whenP is diagonally similar to a symmetric matrix, and several other equivalent conditions for diagonal symmetrizability ofP are collected together here. Other conditions which arise involve further variations upon symmetry, and may be viewed as algebraic descriptions of various features of symmetry. A tool of interest is a slight variation upon a recent characterization of the Perron root.

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Authors and Affiliations

  1. Mathematics Department, College of William and Mary, 23185, Williamsburg, Virginia, USA

    Charles R. Johnson

  2. Departamento de Mathemática, Universidade de Lisboa, 1300, Lisboa, Portugal

    J. A. Dias da Silva

Authors
  1. Charles R. Johnson
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  2. J. A. Dias da Silva
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Additional information

The work of C. R. Johnson was supported in part by National Science Foundation Grant DMS 87-13762 and by Office of Naval Research Contract N00014-87 K-0661. The work of J. A. Dias da Silva was done in conjunction with the Centro de Algebra da Universidade de Lisboa and was carried out while visiting the first author.

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Johnson, C.R., da Silva, J.A.D. Symmetric matrices associated with a nonnegative matrix. Circuits Systems and Signal Process 9, 171–180 (1990). https://doi.org/10.1007/BF01236450

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  • DOI: https://doi.org/10.1007/BF01236450

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