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Some Comments and a Bibliography on the Frucht—Kantorovich and Wielandt Inequalities

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Innovations in Multivariate Statistical Analysis

Part of the book series: Advanced Studies in Theoretical and Applied Econometrics ((ASTA,volume 36))

Abstract

The well-known inequality

$$\frac{{x'Ax \cdot x'{{A}^{{ - 1}}}x}}{{{{{\left( {x'x} \right)}}^{2}}}}\frac{{{{{\left( {{{\lambda }_{{\text{1}}}}{\text{ + }}{{\lambda }_{n}}} \right)}}^{2}}}}{{4{{\lambda }_{{\text{1}}}}{{\lambda }_{n}}}} $$
(1)

is usually attributed (cf. e.g., [35], [36]) to Nobel Laureate and Academician Leonid Vitalévich Kantorovich (1912–1986) for the inequality he established in 1948 in a long survey article (in Russian) on “Functional Analysis and Applied Mathematics” ([111], pp. 142–144; [112], pp. 106–107). In (1) x is a real n х 1 vector and A is a real n х n symmetric positive definite matrix, with λ1 and λn, respectively, its (fixed) largest and smallest, necessarily positive, eigenvalues. (All matrices and vectors in this paper are real.)

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Authors
  1. Gülhan Alpargu
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  2. George P. H. Styan
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Editors and Affiliations

  1. Department of Econometrics, University of Amsterdam, The Netherlands

    R. D. H. Heijmans

  2. Department of Economics, Queen Mary College, University of London, UK

    D. S. G. Pollock

  3. Departament d’Economia i Empresa, Universitat Pompeu Fabra, Spain

    A. Satorra

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Alpargu, G., Styan, G.P.H. (2000). Some Comments and a Bibliography on the Frucht—Kantorovich and Wielandt Inequalities. In: Heijmans, R.D.H., Pollock, D.S.G., Satorra, A. (eds) Innovations in Multivariate Statistical Analysis. Advanced Studies in Theoretical and Applied Econometrics, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4603-0_1

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