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  • 1 Institute of Mathematics, Silesian University of Katowice, 40-007 Katowice, Bankowa Street 14, Poland
  • 2 Institute of Computer Science, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland
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We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

  • [1]

    Belton, A. C. R., Operator Theory, Fourth-year undergraduate course, Lancaster University 2018.

  • [2]

    Bernau, S. J., The square root of a positive self-adjoint operator, J. Austr. Math. Soc., 8 (1968), 1736.

  • [3]

    Bishop, Ch. M., Pattern recognition and machine learning, Springer Verlag 2006.

  • [4]

    Fisher, R., The use of multiple measurements in taxonomic problems, Annals of Eugenics, 7 (1936), 179188.

  • [5]

    Fukunaga, K., Introduction to statistical pattern recognition, Academic Press 1990.

  • [6]

    Krzysko, M., Fundamentals of multivariate statistical inference, Mickiewicz University Publishing House, Poznań 2009.

  • [7]

    Mardia, K. V., Kent, J. T. and Bibby, J. M., Multivariate Analysis, Academic Press Inc, London 1979.

  • [8]

    Pečarić, J. E., Puntanen, S. and Styan, G. P. H., Some further matrix extensions of the Cauchy–Schwarz and Kantorovich inequalities, with some statistical applications, Linear Algebra Appl., 237/238 (1996)

    • Search Google Scholar
    • Export Citation
  • [9]

    Riesz, F. and Sz˝okefalvi-Nagy. B., Functional Analysis, Ungar, New York 1955.

  • [10]

    Schott, J. R., Matrix analysis for statistics, John Wiley & Sons, New Jersey 2017.

  • [11]

    Sebestyén, Z. and Tarcsay, Zs., On the square root of a positive selfadjoint operator, Period. Math. Hungarica, 75 (2017), 268272.

  • [12]

    Steele, M. J., The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities, Cambridge University Press 2004.

    • Search Google Scholar
    • Export Citation
  • [13]

    Stone, M. H., Linear Transformations in Hilbert Spaces and their Applications to Analysis, Amer. Math. Soc. Colloq. Publ., 15, Amer. Math. Soc. 1932.

    • Search Google Scholar
    • Export Citation
  • [14]

    Wouk, A., A note on square roots of positive operators, SIAM Rev., 8 (1966), 100102.

  • [15]

    Wouk, A., A course of Applied Functional Analysis, Wiley, New York 1979.