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Determinant Inequalities with Applications to Isoperimatrical Inequalities

  • Alexander G. Reznikov
Conference paper
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

Let A = (a ij ) be positive definite Hermitian n × n matrix. We prove a following strengthening of the Hadamard inequality:
$$ \frac{{\det \,A}}{{{a_{11}} \cdot \cdot \cdot {a_{nn}}}} \leqslant \exp \left( {\frac{{\sum {_{i < j}{{\left| {{a_{ij}}} \right|}^2}} }}{{n\,{{\max }_{1 \leqslant i \leqslant n\,}}{{\left| {{a_{ij}}} \right|}^2}}}} \right) $$
We give similar estimate in the case of non-Hermitian matrix. We use these results for a short proof of the existence of Von Koh’s infinite determinants, and also give a strong isoperimetric inequality for simplices in ℝn.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Alexander G. Reznikov
    • 1
  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael

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