Skip to main content
SpringerLink
Log in

Extensions of the Hadamard determinant theorem

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Denote byH n the set ofn byn, positive definite hermitian matrices. Hadamard proved thath(A)≧det(A) for allA∈H n, whereh(A) is the product of the main diagonal elements ofA. Subsequently, M. Marcus showed that per(A)≧h(A) for allA∈H n. This article contains a result for all generalized matrix functions from which it follows thath(A)≧(per(A1/n))n,A∈H n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. R. Johnson,The many proofs of Hadamard’s determinantal inequality for positive definite matrices, manuscript.

  2. M. Marcus,The permanent analogue of the Hadamard determinant theorem, Bull. Am. Math. Soc.69 (1963), 494–496.

    MATH  Google Scholar 

  3. M. Marcus,The Hadamard theorem for permanents, Proc. Am. Math. Soc.15 (1964), 967–973.

    Article  MATH  Google Scholar 

  4. M. Marcus and M. Newman,Inequalities for the permanent function, Ann. of Math.75 (1962), 47–62.

    Article  MathSciNet  Google Scholar 

  5. A. W. Marshall and I. Olkin,A convexity proof of Hadamard’s inequality, Am. Math. Monthly89 (1982), 687–688.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Merris,Two problems involving Schur functions, Linear Algebra & Appl.10 (1975), 155–162.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Merris,Extensions of the Minkowski determinant theorem, Port. Math.38 (1979), 149–153.

    MATH  MathSciNet  Google Scholar 

  8. H. Minc,Permanents, inEncyclopedia of Math. and its Applications, Vol. 6, Addison-Wesley, Reading, 1978.

    Google Scholar 

  9. P. J. Nikolai,Mean value and limit theorems for generalized matrix functions, Canad. J. Math.21 (1969), 982–991.

    MATH  MathSciNet  Google Scholar 

  10. I. Schur,Über endliche Gruppen und Hermitesche Formen, Math. Z.1 (1918), 184–207.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, California State University, 94542, Hayward, CA, USA

    Russell Merris

Authors
  1. Russell Merris
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Merris, R. Extensions of the Hadamard determinant theorem. Israel J. Math. 46, 301–304 (1983). https://doi.org/10.1007/BF02762889

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02762889

Keywords

Search

Navigation