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)


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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  1. [1]
    I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear non Self Adjoint Operators, Translation of Mathematical Monographs 18, AMS, Providence, 1968.Google Scholar
  2. [2]
    R. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, N.Y., 1985.zbMATHGoogle Scholar
  3. [3]
    C.R. Johnson, Optimization, Matrix Inequalities and Matrix Completions in J.W. Melton, Operator Theory, Analytic Functions, Matrices and Electrical Engineering, Conference board of the mathematical sciences, 68, AMS, 1987.Google Scholar
  4. [4]
    C.R. Johnson, T. Markham, Compression and Hadamard power inequalities for M-matrices, Linear and Multilinear Algebra 18: 23–34 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    H. von Koch, Sur quelques points de la théorie des déterminants infinis, Acta Math 24: 89–122 (1900).zbMATHCrossRefGoogle Scholar
  6. [6]
    H. von Koch, Sur la convergence des déterminants infinis, Rend. Cire. Mat. Palermo 28: 255–266 (1909).CrossRefGoogle Scholar
  7. [7]
    A. Perelson, Spectral representation of a generalized trace and determinant, Int. Eq. and Op. Theory, 9: (1986).Google Scholar
  8. [8]
    A. Perelson, Generalized Traces and Determinants for Compact Operators, Ph.D. Thesis, University of Tel Aviv, 1987.Google Scholar
  9. [9]
    A.G. Reznikov, A strengthened isoperimetrical inequality for simplices, GAFA seminars, Lecture Notes in Math. 1469 (1991), 90–93.MathSciNetCrossRefGoogle Scholar
  10. [10]
    H. Wolkowitz, G. Styan, Bounds for eigenvalues using traces, Linear Algebra Appi. 29: 471–506 (1980).CrossRefGoogle Scholar

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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