Some results on the structure and spectra of matrix-products

Murad Banaji, Carrie Rutherford

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Abstract

We consider certain matrix-products where successive matrices in the product belong alternately to a particular qualitative class or its transpose. The main theorems relate structural and spectral properties of these matrix-products to the structure of underlying bipartite graphs. One consequence is a characterisation of caterpillars: a graph is a caterpillar if and only if all matrix-products associated with it have real nonnegative spectrum. Several other equivalences of this kind are proved. The work is inspired by certain questions in dynamical systems where such products arise naturally as Jacobian matrices, and the results have implications for the existence and stability of equilibria in these systems.
Original languageEnglish
Pages (from-to)192-212
JournalLinear Algebra and its Applications
Volume474
Early online date6 Mar 2015
DOIs
Publication statusPublished - 1 Jun 2015

Keywords

  • trees
  • caterpillars
  • P-matrices
  • matrix spectra

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