Eigenvalue Optimization of Polynomials in Non-commuting Variables

  • Sabine Burgdorf
  • Igor Klep
  • Janez Povh
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In Sect. 1.6 we introduced a natural notion of positivity that corresponds exactly to nc polynomials that are SOHS. Recall that an nc polynomial is positive semidefinite if it yields a positive semidefinite matrix when we replace the letters (variables) in the polynomial by symmetric matrices of the same order. Helton’s Theorem 1.30 implies that positive semidefinite polynomials are exactly the SOHS polynomials, the set of which we denoted by Σ2.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sabine Burgdorf
    • 1
  • Igor Klep
    • 2
  • Janez Povh
    • 3
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Department of MathematicsThe University of AucklandAucklandNew Zealand
  3. 3.Faculty of Information Studies in Novo MestoNovo MestoSlovenia

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