Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

Michal Gnacik, Tomasz Kania

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Abstract

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever λ2,…,λn are non-positive real numbers with 1 + λ+…+ λn ⩾ 1/2, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely (1, λ2,…,λn). We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.
Original languageEnglish
Pages (from-to)175-187
JournalLinear Algebra and its Applications
Volume592
Early online date23 Jan 2020
DOIs
Publication statusPublished - 1 May 2020

Keywords

  • doubly stochastic matrix
  • bistochastic matrix
  • inverse problem
  • SDIEP
  • Suleĭmanova spectrum

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